An Introduction To Dynamical Systems

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Transcript of An Introduction To Dynamical Systems
T o VLADIMIR IGOREVICH ARNOLD
and STEPHEN SMALE
for their inspirational work
An introduction to
D Y N A M I C A L S Y S T E M S
D. K. ARROWSMITH Lec~urer, School of Ma~hematical Sciences,
Queen Mary & Westfiild College, Uniwsity o/ London
C. M. PLACE Lecturer (formerly Department of Marhemarics,
Westjield College, Uniwrsiry of London)
I I 1 1
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2 6 3 ~ 5 4 c I E
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)(:I35 % Y: 91 Z CAMBRIDGE UNIVERSITY PRESS
CONTENTS
Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 IRP
40 West 20th Street, New York, NY 100114211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia
8 Cambridge University Press, 1990
First published 1990 Reprinted 1991, 1994
Pi~nted in Great Britain by Bell and Bain Ltd, Glasgow
British Library cataloguing in publication data Arrowsmith, D.K.
An introduction to dynamical systems. 1. DilTerentiable dynamical systems 1. Title 11. Place, C.M. 514.7
Library of Congress cataloguing in publication data Arrowsmith, D. K.
An introduction to dynamics1 systems 1 D.K. Arrowsmith and C.M. Plaa p. Cm.
Bibliography: p. Includes index. ISBN 0 521 30362 1.  ISBN 0 521 31650 2 (paperback) 1. Diflercntiable dynamicsl systems. I. Plaa, C. M. 11. Title.
QA614.8.A77 1990 515.'.3524~20 897191 CIP
Transferred to digital reprinting 2001 Printed in the United States of America
1 DiR ad flow, 1 .l introduction 1.2 Elementary dynamics of dikomorphisms
1.2.1 Definitions 1.2.2 Diffeomorphisms of the circle
1.3 Flows and differential equations 1.4 Invariant sets 1.5 Conjugacy 1.6 Equivalence of flows 1.7 PoincarC maps and suspensions 1.8 Periodic nonautonomous systems 1.9 Hamiltonian flows and PoincarC maps
Exercises
2 Local properties of flows and diffeomorphisms 2.1 Hyperbolic linear diffeomorphisms and flows 2.2 Hyperbolic nonlinear fixed points
2.2.1 Diffeomorphisms 2.2.2 Flows
2.3 Normal forms for vector fields 2.4 Nonhyperbolic singular points of vector fields 2.5 Normal forms for diffeomorphisms 2.6 Timedependent nonnal forms 2.7 Centre manifolds 2.8 Blowingup techniques on R2
2.8.1 Polar blowingup 2.8.2 Directional blowingup Exercises
3 Shctwal stability, hyperblieity a d Lolaoelinic paiob ' 3.1 Structural stability of linear systems 3.2 Local structural stability 3.3 Flows on twodimensional manifolds 3.4 Anosov diffeomorphisms
Contents Contents
3.5 Horseshoe difleomorphisms 3.5.1 The canonical example 3.5.2 Dynamics on symbol sequences 3.5.3 Symbolic dynamics for the horseshoe diffeomorphism
3.6 Hyperbolic structure and basic sets 3.7 Homoclinic points 3.8 The Melnikov function
Exercises
4 Local bifurcations I: planar vector fields and diffeomorphisms on R
4.1 lntroduction 4.2 Saddlenode and Hopf bifurcations
4.2.1 Saddlenode bifurcation 4.2.2 Hopf bifurcation
4.3 Cusp and generalised Hopf bifurcations 4.3.1 Cusp bifurcation 4.3.2 Generalised Hopf bifurcations
4.4 DiNeomorphisms on R 4.4.1 D,f(O) = + I: the fold bifurcation 4.4.2 D,f(O) =  1 : the flip bifurcation
4.5 The logistic map Exercises
5 Locnl bifurcations 11: diffeomorphisms on W2 5.1 lntroduction 5.2 Arnold's circle map 5.3 Irrational rotations 5.4 Rational rotations and weak resonance 5.5 Vector field approximations
5.5.1 Irrational fl 5.5.2 Rational /I = p/q, q >, 3 5.5.3 Rational $= pfq. q = 1,2
5.6 Equivariant versa1 unfoldings for vector field approximations 5.6.1 q = 2 5.6.2 q = 3 5.6.3 q = 4 5.6.4 q >, 5
5.7 Unfoldings of rotations and shears Exercises
6 Areapreserving maps and their perturbations 6.1 lntroduction 6.2 Rational rotation numbers and BirkhoN periodic points
6.2.1 The PoincarbBirkhoB Theorem 6.2.2 Vector field approximations and island chains
6.3 Irrational rotation numbers and the KAM Theorem 6.4 The AubryMather Theorem
6.4.1 Invariant Cantor sets for homeomorphisms on S' 6.4.2 Twist homeomorphisms and Mather sets
6.5 Generic elliptic points 6.6 Weakly dissipative systems and Birkhoff attractors
BirkhoN periodic orbits and Hopf bifurcations Double invariant circle bifurcations in planar maps Exercises
Hi for e x a d a s  References Index
P R E F A C E DifJ^eomorphisms and flows
In recent years there has been a marked increase of research interest in dynamical systems and a number of excellent postgraduate texts have been published. This book is specifically aimed at the interface between undergraduate and postgraduate studies. It is intended both to stimulate the interest of final year undergraduates and to provide a solid foundation for postgraduates who intend to embark on research in the field. For example, a challenging thirdyear undergraduate course can be constructed by selecting topics from the first four chapters. Indeed, lecture courses taught by one of us (CMP) provided the basis for Chapters 1, 2 and 4. On the other hand, Chapter 6 is directed at firstyear postgraduate students. It contains a selection of current research topics that illustrate the interaction between superficially different research problems.
A major feature of the book is its extensive set of exercises; more than 300 in all. These exercises not only illustrate the topics discussed in the text, but also guide the reader in the completion of technical details omitted from the main discussion. Detailed model solutions have been prepared and hints to their construction are provided.
The reader is assumed to have attended courses in analysis and linear algebra to secondyear undergraduate standard. Prior knowledge of dynamical systems is not necessary; however, some familiarity with the qualitative theory of differential equations and Hamiltonian dynamics might be an advantage.
We would like to thank Martin Casdagli for sharpening our understanding of Birkhoff attractors, David Knowles and Chris Norman for helpful discussions and Carl Murray for steering some awkward diagrams to a laser printer. We are grateful to the Quarterly Journal oj' Applied Mathematics and SpringerVerlag for allowing us to use diagrams from some of their publications and our thanks go to Sandra Place for her fast and accurate typing of much of the manuscript. One of us (CMP) would like to'acknowledge the Brayshay Foundation for its financial support throughout this project. Finally, we must both pay tribute to the patience and support of our families during the long, and often difficult, gestation period of the manuscript.
1.1 Introduction
A dynamical system is one whose state changes with time ( t ) . Two main types of dynamical system are encountered in applications: those for which the time variable is discrete ( t ~ Z or N) and those for which it is continuous (t E R).
Discrete dynamical systems can be presented as the iteration of a function, i.e.
x,+,=f(x,), t ~ z o r b l . (1.1.1)
When t is continuous, the dynamics are usually described by a differential equation
In (1.1.1 and 2), x represents the state of the system and takes values in the state or phase space. Sometimes the phase space is Euclidean space or a subset thereof, but it can also be a nonEuclidean structure such as a circle, a sphere, a torus or some other d~gerentiable manifold.
In this chapter we will consider two special cases of the above equations, namely when:
(i) f i n (1.1.1) is a dgeomorphism; and
(ii) the solutions of (1.1.2) can be described by a Jlow with velocity given by the vector field X.
These two cases have been widely studied and they are fundamental to our understanding of dynamical systems. Smale, in his definitive work (Smale, 1967), pointed out that (i) and (ii) are closely related and our discussion emphasises this connection.
Any description of the theory of (i) and (ii) involves differentiable maps so let us begin by recalling some definitions. Let U be an open subset of Rn. Then a function g: U + R is said to be of class Cr if it is rfold continuously differentiable, 1 < r < a. Let V be an open subset of Rm and G: U + V. Given coordinates
2 1 Diffeomorphisn~s and Jows 1.1 Introduction 3
(x,, . . ., s,) in U and (y,, . . ., y,) in V, G may be expressed in terms of component functions gi: U + R, where
The map G is called a Crmap if gi is C' for each i = 1,. . ., m. G is said to be diflerentiable if it is a Crmap for some 1 < r ,< oo and to be smooth if it is Cm. Maps that are continuous but not differentiable are, conventionally, referred to as Comaps.
Definition 1.1.1 G is said to be a diffeomorphism if it is a bijection and both G and G' are differentiable mappings. G is called a Ckdiffeomorphism if both G and G  ' are Ckmaps.
Observe that the bijection G: U + V is a diffeomorphism if and only if m = n and the matrix of partial derivatives
is nonsingular at every x E U. Thus G(x, y) = (exp(y), exp(x)f with U = RZ and V = {(x, y)lx, y > 0) is a diffeomorphism because Det DG(x, y) = exp(x + y) # 0 for each (x, y ) ~ RZ.
If G satisfies Definition 1.1.1 with G and G ' continuous, rather than differentiable, maps then G is said to be a homeomorphism. As we shall see, such maps play a central role in the topological theory of flows and diffeomorphisms.
The above definitions are adequate provided phase space is Euclidean, but, as we have already mentioned, the natural setting for dynamics is a diflerentiable manijold. The important point here is that manifolds have the property that they are 'locally Euclidean' and this allows us to extend the idea of differentiability to functions defined on them. If M is a manifold of dimension n then, for any x E M, there is a neighbourhood W c M containing x and a homeomorphism h: W + Rn which maps W onto a neighbourhood of h ( x ) ~ 88". Since we can define coordinates in U = h(W) G Rn (the coordinate curves of which can be mapped back onto W), we can think of h as defining local coordinates on the patch W of M (see Figure 1.1).
The pair (U, h) is called a chart and we can use it to give meaning to differentiability on W. Let us assume, for simplicity, that f: W + W, then f induces a map ?= hf.h': U + U (see Figure 1.2). We say that f is a 6  m a p on W if? is a Ckmap on U. This construction allows us to give a definition of a local diffeomorphism on M.
In order to obtain a global description of the manifold, we cover it with a family of open sets, W,, each with its associated chart (U,, ha) (predictably, the set of all charts is called an atlas). If Wan Wp is not empty, then either (U,, ha) or (Up, hb) can be used to provide local coordinates for Wan Wp. This possibility induces overlap maps, hap and hpa between ha(Wan Wp) c Ua and hp(Wan Wp) G Up (see
Figure 1.1 Examples of differentiable manifolds and some 'patches' of local coordinates. Several open sets based on patches of this kind may be required in order to cover the whole manifold.
I (c) sphere : polar 0 e
coordinates
(d) sphere : stereographic projection
.F
4 I Diffeomorphisms and jlows 1.2 Elementary dynamics of diffeomorphisms 5
Figure 1.3). If we now consider f: Wan W, + Wan W,, we have  two alternative . representatives?, = ha. f . h i and?, = h,.f. ha ' for f. ~ i n c e ? ~ and f, are determined by different charts, they might belong to different differentiability classes, so that the class off would be ambiguous. A manifold is said to be diflerentiable if all the overlap maps are diffeomorphisms of the same differentiability class, C' say. Now, from Figure 1.3,
= hP.f.hi1
=(h,.h,').(h,.f.h,').(h,.h,')
= h,,.?,.h,'. (1.1.5)
Thus all local representatives off have the same differentiability class, Ck say, with k < r. It is important to note that r is determined entirely by the charts and hence by the structure of M. A manifold with overlap maps of class C' is called a Cmanifold.
The discussion presented above is, of course, incomplete. We have only considered maps taking a chart into itself. This is clearly not true in general. Given f: M t M, then f: W, + Wg and f: Wan W, + Wpn W,. The generalisation of our simple arguments that allows for these omissions is considered in Exercise 1.1.2. Needless to say, the 'message' is unchanged by these manipulations.
A more detailed discussion of difirentiable manifolds is not necessary here (the interested reader should consult Arnold (1973) or Chillingworth (1976)). While the ideas outlined above provide valuable background knowledge, we will rarely find ourselves involved with charts, atlases, etc. This is because our concern is the dynamics of maps defined on M given that they are diffeomorphisms or flows.
Figure 1.2 Commutative diagram illustrating the representation of f defined on an open set W of M in a local shart (U, h).
Figure 1.3 Illustration of the definition of the overlap maps h,! and hb,. Note that hp, = h&'.
These maps are usually presented to us in local coordinates so that the manifold structure does not appear explicitly.
1.2 Elementary dynamics of diffeomorphisms
1.2.1 Definitions
Let M be a differentiable manifold and suppose f: M + M is a diffeomorphism. For each X E M, the iteration (1.1.1) generates a sequence, the distinct points of which define the orbit or trajectory of x under f. More precisely, the orbit of x under f is { P ( x ) l m ~ Z}. For m~ Z +, F is the composition off with itself m times. Since f is a diffeomorphism f ' exists and f" = (f l)". Finally, f" = id,, the identity map on M. Typically, the orbit of x is a biinfinite sequence of distinct points of M. However, there are two important exceptions to this state of affairs.
Definition 1.2.1 A point x* E M is calleda fixed point off if fm(x*) = x* for allm EZ.
Definition 1.2.2 A point X* E M is a periodic point of f if P(x*) = x*, for some integer q 2 1.
The least value of q satisfying Definition 1.2.2 is called the period of the point x* and the orbit of x*, i.e.
is said to be a periodic orbit of period q or a qcycle off. Clearly, since fq(x*) = x*, it is the sequence {F(x*)),", , which is qperiodic. Notice that a fixed point is a periodic point of period one and a periodic point off with period q is a fixed point of P. Morever, if x* is a periodic point of period q for f then so are ail of the other points in the orbit of x*. For example, if fq(x*) = x* then f(P(x*)) = f(x*) = fq(f(x*)) and f(x*) is therefore a periodic point of period q, and so on for f2(x*), . . ., fqyx*).
Fixed and periodic points can be classified according to the behaviour of the orbits of points in their vicinity. The following ideas are due to Liapunov.
Definition 1.2.3 A fixed point, x*, is said to be stable i f , for every neighbourhood N of x*, there is a neighbourhood N' E N of x* such that if x E N' then F(x)E N for all m > 0.
Essentially, Definition 1.2.3 says that iterates of points 'near to' a stable fixed point, remain 'near to' it for m~ E +. If a fixed point x* is stable and Lim r ( x ) = x*,
m+m
for all x in some neighbourhood of x*, then the fixed point is said to be asymptotically stable. Trajectories of points near to an asymptotically stable fixed point move toward it as m increases. Fixed points that are stable, but not
6 1 Diffeornorphisms and flows 1.2 Elementary dynamics of d~yeomorphisms 7
asymptotically stable, are said to be neutrally or marginally stable and those that are not stable in the sense of Definition 1.2.3 are unstable.
1.2.2 Dijfeomorphisms of the circle
The circle (S1) is arguably the simplest nonEuclidean differentiable manifold. It is compact (see Chillingworth, 1976, p. 143) so 'behaviour at infinity' is not a problem; it has no boundary so that dynamics can be studied without the complication of boundary conditions on the functions concerned and it is onedimensional. The dynamics of diffeomorphisms on the circle therefore provide an ideal opportunity for us to illustrate the definitions given in 4 1.2.1.
Some of the simplest examples of diffeomorphisms on S1 are the pure rotations. They are easily defined in terms of the angular displacement (8) at the centre of the circle relative to a reference radius (see Figure 1.4). In terms of this local coordinate, an anticlockwise rotation by a may be written as
R,(8) = (8 + a) mod 1. (1.2.2)
Here we have assumed that 0 is measured in units of 271. If a = p/q, p, ~ E E and relatively prime, then
R:(0) = (0 + p) mod l = 0 (1.2.3)
and we conclude (cf. Definition 1.2.2) that every point of the circle is a periodic point of periodq, i.e. the orbit of any point is a qcycle (see Figure 1.4). If a is irrational then
RF(0) = (0 + ma) mod 1 # 9, (1.2.4)
for any 8 and, in fact, the orbit of any point fills the circle densely (see Exercise 1.2.1 ). Obviously more general diffeomorphisms of S' do not simply rotate all points
uniformly. Crudely speaking they compress some arcs of the circle and stretch
Figure 1.4 Typical orbit of the pure rotation R, for a = p/q = 215. Observe that the orbit of 0 winds around the circle p = 2 times before returning to 0 on the lifth iteration.
others. It is then dificult to recognise fixed or periodic points from the representation of orbits on the circle itself. This is a problem for any map ( f ) of the circle, whether it is a diffeomorphism or not, and it is solved by considering a lift off.
The natural setting for introducing the lift of f: S1 + S1 is when f is a homeomorphism rather than a diffeomorphism and it would be perverse to artificially confine our discussion to the differentiable case. Moreover, by taking f to be a homeomorphism at this point we can better appreciate the consequences of imposing differentiability on f and f  I . Thus, let f: S1 , S1 be a homeomorphism and suppose there is a continuous functionf: R + R such that
(see Figure IS), where
n(x) = x mod 1 = 8. (1.2.6)
hen f is called a lift of J: S1 + S' onto W.
Proposition 1.2.1 Let 7 be a l i f t of the orientationpreserving homeomorphism f: S1 + S'. Then
for every x c R.
Proof. Observe that
because n(x) = n(x + 1) by (1.2.6). If we substitute for f . n from (1.2.5), (1.2.8) becomes
n t f b ) ) = n(f(x + 1))
and it follows that
where k(x) is an integer possibly depending on x. However, since 7 is continuous, k(x) must be continuous and this is only possible if k(x) = ~ E Z .
Figure 1.5 Commutative diagram illustrating the definition of the lift of a circle homeomorphism f. The map n takes infinitely many equivalent points of R onto a single point of S'.
8 1 Difiomorphisms and Jlows 1.2 Elementary dynamics of [email protected] 9
Suppose k > 2, then ~f(x) and f(x + I ) differ by more than two and 7 takes the form shown schematically in Figure 1.6(a). Clearly, the points x, and x, satisfying f ix , ) = 1 and fix,) = 2 are both less than unity. This means that x maps them to distinct points on S1. However, f(x,) and f(xl) differ by unity and therefore represent the same point on S'. This contradicts the hypothesis that f is a homeomorphism. Hence k < I .
If k = 0, f(0) =f(1) and f fails to be injective on (0 , l ) (see Figure 1.6(b)). Again this contradicts the fact that f is a homeomorphism.
If k < 0 then continuity of f can only be maintained iff is orientationreversing in contradiction to hypothesis. Moreover, it is clear that similar arguments would lead to a minus sign in the right hand side of (1.2.7) for orientationreversing f.
Finally, we conclude that k = 1 and (1.2.7) follows. 0
It is important to realise that not every continuous function satisfying (1.2.7) is the lift of some homeomorphism. The function shown in Figure 1.7 is continuous and satisfies (1.2.7) but fails to be the lift of a homeomorphism because it is not injective. Figure 1.7 also highlights the geometrical significance of (1.2.7); namely that the graph off in the interval [k, k + 11 is obtained by shifting the graph of 7 in [0, 11 vertically by k units. In this way any continuous function g, defined on [0, 11, that is injective, and such that g(l) =g(O) + 1, can be used to construct a lift f for some homeomorphism f: S1 4 S1. The function f is given by (1.2.5). A simple example of this construction is given in Figure 1.8(a) where
g(x)= .xZ + 2 x + & (1.2.1 1)
x€[O, I]. In this case, f is a continuous bijection but it is not differentiable at x = l , 2 , . . . . This reflects on the corresponding f which is a homeomorphism,
Figure 1.6 Schematic forms for f when (1.2.10) has (a) k = 2; (b ) k = 0. In both cases, the hypothesis that fisa homeomorphism iscontradicted.
but not a diffeomorphism of S ' . To obtain the latter, f must be a bijection and differentiable for all x E OW. An example of this type is shown in Figure 1.8(b) where
g(x) = x + + + &sin 2nx, (1.2.12)
x€[O, 11. Notice, we have, without loss of generality, taken f (0)~[0 ,1) in both of the
above examples. Observe that, n(f(x) + k) = n(f(x)), for any k E Z. Thus if T(x) is a lift o f f then so is J,(x) =fix) + k, k E Z. Therefore, unless otherwise stated, we will assume that 7 is the member of this family of lifts satisfying T(o)E[O, 1).
Figure 1.7 The function 7 shown here cannot be the lift of a homeo morphism f: St + SL because it is not injective.
Figure 1.8 The function shown in (a) is the lift of a homeomorphism, but not of a diffeomorphism, of the circle. Lifts of diffeomorphisms are diflerentiable functions of x, see (b) for example, where 7 is obtained from (1.2.12).
10 1 DifSeomorphisrns and flows 1.3 Flows and differential equations 11
How are the fixed or periodic points of f : S1 + S1 related to the properties of the lift 7?
Proposition 1.2.2 Let f: S1 + S1 be an orientationpreserving homeomorphism and suppose that f is the lijl of f with f ( 0 ) ~ [0, 1). Then n(x*) is a fixed point off $ and only i f either
J(x*) = .x* (1.2.13a)
Proof. If fix*) = x* (or ,fix*) = x* + 1) then
n(f(x*)) = n(x*) (or n(f(x*)) = n(x* + 1) = n(x*)). (1.2.14)
In either case,
f(n(xS)) = n(x*)
by (1.2.5) and n(x*) is a fixed point off. If O* = x(x*) is a fixed point off, i.e. f(O*) = 0*, then
f(n(x*)) = n(x*) = x(~(x*)) (1.2.16)
by (1.2.5). Thus
f(x*)=x*+k, keZ. (1.2.17)
Let x*=y*+l , IEZ, y*€[O, 1) then (1.2.17) becomes
f ( y * ) + ~ = y * + ~ + k . (1.2.18)
Here we have noted that a simple induction on J(x + I) =f(x) + 1 gives S(x + 1) = f(x) + 1. Thus, if (1.2.17) is satisfied for any x*, it must be satisfied for a point y* E [0, I). Now, f(1) = f(0) + 1 and Jis injective so that 3 0 ) <fly) < fi0) + 1 for Y E [0, 1). Therefore, (1.2.18) cannot be satisfied unless k = 0 or 1 (see Figure 1.9).
Proposition (1.2.2) can be used to locate periodic points off. Suppose that f has lift 7, i.e. n(f(x)) = f(n(x)), XER, then
n(f '(x)) = n(f(f(x))) =f(ntJ(x)) = J2(n(x)). (1.2.19)
Thus f 2 is a lift of f 2.. It only remains to ensure that , ~ ( o ) E [0, 1) (i.e. choose the lift ,i2  [fZ(0)], where [.I denotes the integer part of ), and Proposition 1.2.2 allows us to find the period2 points off . These arguments obviously extend to points of period q > 2.
An alternative approach is to recognise that if f ' ( ~ ) ~ [ i , 1 + 1) then (1.2.13) is
replaced by
 This point of view often has the advantage that f, f ', . . ., f ', . . . can be presented on the same diagram (see Figure 1.10) without ending up with a confusion of curves in the vicinity of y = x and y = x + 1.
The lift f of f: S1 + S' not only provides a means of conveniently finding fixed and periodic points, it can also allow us to determine their stability. If (1.2.13a) is satisfied at x*, then the orbits of points near to x* under f can be obtained by moving between y =f(x) and y = x as in Figure 1.1 1. The fixed point x* is stable (unstable) if
(see any first course in Numerical Analysis). The stability of B* = n(x*) is clearly the same as that of x*. When (1.2.13b) is satisfied, we can either replace f by f  I, so that x* is then represented by an intersection with y = x, and proceed as above or construct paths for the orbits of 7 by using y = 7(x) and y = x t 1 . The stability of the fixed point is still given by (1.2.21).
1.3 Flows and differential equations
The iteration problem (I. 1 .l) for a diffeomorphism f : M + M given different X,E M is equivalent to the study of the set of functions { P ] m E Z). This set has the property
Figure 1.9 Examples illustrating why (1.2.18) can only be satisfied if k = 0 or 1 for the case when f is an orientationpreserving homeo morphism. As f, shows, if k > 1 then /fails to be injective. Notice (1.2.17) has a countable infinity of solutions for each solution to (1.2.18).
12 I Diffeomorphisms and flows 1.3 Flows and digerential equations 13
that:
f" = i d , and f'.f' = f'+j, (1.3.1)
for each i , j cZ . It is said to be an action of the group Z on M or, more precisely, the Eaction generated by f (see Chillingworth, 1976). In this section we consider the action of the group R on M; such Ractions are called flows on M.
Definition 1.3.1 Aflowon M isacontinuouslyd~$erentiable function Q: R x M + M such that, for each t~ W, the restriction cp(t, .) = Q,(.) satisfies
(a) Q,, =id,; (1.3.2a)
(b) cpl~cps=cpl+S, ~ , s E R . (1.3.2b)
Observe that (1.3.2a and b) imply that (9,)I exists and is given by Q,. Since cp E C1, it follows (see Exercise 1.3.1) that Q,: M + M is a diffeomorphism for each t € R.
Let us pursue the analogy with difleomorphisms a little further. We define the orbit or trajectory of Q through x to be {(p,(x)lt~R) oriented in the sense of
Figufe 1.10 Plots of J'(x) vs x for (a) /(x)= x2+2x + f ; (b) f(x) = x + $ + & sin 2xx. Observe that case (a) corresponds to a homeomorphism with a 3cycle but no fixed points or Zcycles. On the other hand, case (b) is the lift of a diffeomorphism with 2, 4 and 6cycles but no I  , 3 or 5cycles.
increasing t . It can be shown (see Exercise 1.3.2) that there is one and only one trajectory of Q passing through each point x E M. If Q,(x*) = x* for all t E 88 then x* is said to be a fixed point of the flow. Fixed points of flows can be stable, asymptotically stable, neutrally stable or unstable in the sense of Liapunov. Precise definitions are obtained by the transcription PI+ Q, and m E Z H t E R in Definition 1.2.3 and the comments following it.
The orbit of a fixed point is just the point itself. If x is not a fixed point it is said to be ordinary or regular. The trajectory through an ordinary point gives rise to an oriented curve on M and Q has periodic points if this curve is closed.
Definition 13.2 A closed orbit of a flow is a trajectory, y , which is not a fixed point but is such that Q,(x) = x for some x E y and 7 # 0.
Clearly, if Q,(x) = x the orbit returns to x after time 7. If T is the least, positive time for which this occurs, x is a periodic point with period T. It is easily shown (see Exercise 1.3.3) that if a closed orbit has one point with period T, then every point of y is periodic with period T. Thus, T is also called the period of y.
The set of all trajectories of a flow is called its phase portrait. Since each trajectory corresponds geometrically to an oriented curve or point on M, a valuable pictorial representation of the flow is obtained by sketching or plotting typical trajectories. Some examples are shown in Figure 1.12. Notice that the caption to this figure does not specify Q,, instead a differential equation is given. How are flows related
Figure 1.11 Graphical illustration of the iteration x,, , =f(x,) showing the stability ofx:, x: + 1,. . . . Note that IDf(x8] i 1 for all these points. The remaining fixed points, xz, xf + 1, . . ., satisfy IDf(x*# > 1 and are unstable. Observe that the graphical representation of the iteration can still give the stability of a fixed point x* even when IDf(x*)l= 1.
14 1 Diffeornorphisms and Bows 1.3 Flows and differential equations 15
to differential equations? We define the velocity or vector jield, X, of a flow Q by
dQ, X ( x ) =  ( x ) J t = , = Lim dt e  o E
for each x E M. Geometrically, { q l ( x ) J t E R) defines a curve on M passing through x . The vector X ( x ) is directed along the tangent to this curve at x and has magnitude equal to the speed of description of the curve under the parametrisation by t . It is important to realise that, in contrast to vector fields defined on W", X ( x ) # M . For each X E M , the set, T M , , of all vectors tangent to M at x is called the tangent space to M at x and X ( X ) E T M , . Figure 1.13 illustrates T M , for a typical point x e S Z . If M is an ndimensional manifold, then TM, is isomorphic to R for all x E M . Each element of TM, corresponds to an equivalence class of curves on M having the same tangent vector at x (see Chillingworth, 1976, p. 164).
Proposition 1.3.1 Q,(x,) is the solution of x = X ( x ) which passes through xo at t = 0.
Proof. Let { ( t ) = q , (xo) . Then
= Lim c  0
= Lim ~ * Q ~ ( x ~ )  Q ~ ( x ~ ) } e0 E
= Lim c 0 1
Figure 1.12 Sqme examples of phase portraits of flows: (a) 0 = z, i = sin 0; (b) 0 = sin 0, @ = 0; (c) 0 = O(0  (3rr/4))(0  n), @ = 0(n  0).
Thus, C(t) is a solution of x = X ( x ) and, since Q, = id,, { ( o ) = Q,(x,) = x,, as required. 0
Notice that if X(x*) = 0 then q l ( x * ) = x* is the solution of x = X ( x ) passing through x*. Moreover, if Q,(x*) = x* for all t then (1.3.3) implies X(x*) = 0. We conclude, therefore, that x* is a fixed point of Q, if and only if X(x*) = 0. Such points are referred to as singular points of the vector field X .
Proposition 1.3.1 means that every flow on M corresponds to an autonomous differential equation. Unfortunately the converse is not true. This is because there are autonomous differential equations with solutions that cannot be extended indefinitely in t. For example, i = xZ has general solution
t E ( a), C ) ;
~ E R ; (1.3.5)
  t E ( C ( , a ) ) ,
C, C'E W. Only the trivial solution has domain R. In such cases, local flows can still be defined. For example, the function
provides a local flow for f = x 2 . When xo > 0, (1.3.5) implies t ~ (  co, x i ' ) in (1.3.6). It is easy to verify that cp, satisfies (1.3.2) provided t, s and t + s all belong to ( co, x i I). The same function cp, can be used when x , < 0 provided t is restricted
Figure 1.13 Illustration of the tangent space, TM,, to the sphere S2 at x. Let the circles a and b define the latitude and longitude of x . If a and b are tangent to a and b, respectively, at x then TM, = Sp{a, b). Observe that b, c and d are all curves on the sphere having tangent vector b.
16 I Diffeomorphisms and ,flows 1.4 Invariant sets 17
to the interval (xi ' , a). Equation (1.3.6) obviously provides the trivial solution when x , = 0. This local flow is suflicient to characterise the solutions of .i = xZ in a neighbourhood of the origin of the r, xplane. For example, for lxol < 6, (1.3.6) certainly gives the solutions to i = x2 for t E (E', &I). Flows of this type are frequently used implicitly when local properties are discussed (e.g. the saddlenode singularity in Example 2.7.4).
With the above proviso in mind, differential equations, vector fields and flows merely provide alternative ways of presenting the same dynamics. These alternatives have arisen for historical reasons; applications frequently lead to differential equations; local analysis is usually presented in terms of vector fields; and global analysis uses the language of flows. We hope the reader will become familiar with all three possibilities.
1.4 Invariant sets
Sometimes the orbit of a point under f or cp remains within a particular region of phase space for all m s Z or r E R. A set A G M is said to be invariant under the diffeomorphism f (or flow cp) if P(x)E A ( q I ( x ) ~ A ) for each X E A and all m c Z (t E Kt). We write
( A ) s A for all m E E (1.4.1)
or
cpf(A) c A for all t E R. (1.4.2)
Invariant sets are said to be positively (rtegatioely) invariant if the orbits of their elements remain within them for m € E t (H) or t 2 0 (t < 0).
Clearly, the orbit of any point is an example of an invariant set. It follows therefore that fixed points, cycles and closed orbits are all invariant sets. However, they are rather special in two main ways.
(i) They are minimal in the sense that they do not have any proper subsets that are themselves invariant. For example, the circle V is an invariant set for both of the flows shown in Figure 1.14. In contrast to the flow shown in (a), the circle V in (b) has proper subsets, P,, P, , T, and T,, that are invariant under the flow.
(ii) They exhibit periodicity. This is particularly important for applications where such sets frequently correspond to observable phenomena.
More subtle forms ofrecurrence than periodicity can occur in dynamical systems and the following definitions allow us to describe them.
Definition 1.4.1 A point x is a nonwandering point for the difleomorphism f (or flow cp) iS, given any neiglthourhood W of x, there e.xists some m > 0 (t > to > 0) .for whicli fm(W)n W (q,(W)n W) is not empty.
The set of nonwandering points for f (cp) is called the nonwandering set, B(f) (B(cp)). It is easy to see that fixed points and periodic orbits lie in B (see Exercises 1.4.2 and 1.4.3), however, points exhibiting milder forms of recurrence are also present. For example, consider an irrational rotation of the circle, S1. No point of the circle is periodic, but the orbit of any point x ultimately approaches x arbitrarily closely. Thus, every point of S' is a nonwandering point and B = S'.
The structure of B will be examined more closely in 43.6, but we can recognise some important subsets of it by formalising the idea that fixed points and closed, orbits frequently attract or repel the trajectories of phase points not contained in them.
Definition 1.4.2 A point EM is said to be an limit point of the {; trajectory of f (Q) through x ij there is a sequence mi (ti) + such that
Lim Pi(x) = y (Lim cpll(x) = y). i + m i m
limit points of x is known as the limit set of x, denoted {: Lab) . These sets are invariant under f ( q ) Let z = P(y), m s Z (z = yf(y),
t E R), where y satisfies Definition 1.4.2. Then Lim P t m ( x ) = z (Lim cp,,,,(x) = z) i +m i+m
so that z and y belong to the same limit set of x. Notice that a and wlimits sets are subsets of B for any x. Recall if y $ B then
Figure 1.14 The circle V is an invariant set for both of the flows shown. However, in (a) V has no proper subsets that are themselves invariant; while in (b) V is the disjoint union of the invariant sets P,, P,, T,, T,.
18 1 Diffeomorphisms and jlows 1.4 Invariant sets 19
there exists a neighbourhood V 3 y such that f"(V)n V is empty for all m > 0. However, y E L,(x) implies F i ( x ) € V for i 2 N, say, and hence there is z = fN(x)e V such that F'"(z)E V for i > N. Thus P ( V ) n V cannot be empty for all m and y must lie in R.
Example 1.4.1 Find L,(x) and L,(x) for (a ) x = 0 ; (b ) x # 0, when Q is the flow on WZ induced by
1 r ) 0 = 1, (1.4.3)
where (r , 9 ) are plane polar coordinates.
Solution. Q has a unique, attracting closed orbit y given by r( t ) r 1 , with period T = 2n, and an unstable fixed point at the origin (see Figure 1 . I S ) .
( R ) X = 0 Note Q,(O) = 0 for all t therefore
L,(O) = L,(O) = ( 0 ) . (1.4.4)
( b ) x # O Let y = (cos go, sin e O ) € y and lei ti be the sequence o f t > 0 at which the orbit of x crosses the radial line from 0 through y. Then Lim cp,,(x) = y and y is an olimit
i  m
point of x. This argument is valid for any y E y and any x # 0 . Therefore, L,(x) = y for any x # 0.
A similar argument allows us to show that
Figure 1.15 Phase portrait for the flow of (1.4.3).
However, for 1x1 > 1 , the Lim Q,,(x) does not exist for any sequence t , such that i  m
ti +  co as i , a, and therefore L,(x) is empty.
Example 1.4.2 Let the flow Q have the phase portrait shown in Figure 1.16. What are L,(x) and L,(x) for x E A, B, C respectively? What feature do all three olimit sets have in common?
Solution. Sequences {t,}?=, can be constructed as in Example 1.4.1 to show that
x E C: L,(x) = empty set; L,(x) = d A u JB.
Let r, and I, be the trajectories of the flow which form the separatrices of the saddle point Po. Observe that
and it follows that all three olimit sets are unions of fixed points and the trajectories joining them.
Example 1.4.2 illustrates an important theorem concerning the global properties of planar flows.
Theorem 1.4.1 (PoincarkBendixson) A nonempty, compact limit set of a Jow on the plane, which contains no fixed point, is a closed orbit.
This theorem states that the types of limit sets illustrated in Examples 1.4.1 and 1.4.2 are the only compact ones that can occur in flows on the plane. It is one of the few theorems which gives the existence of a global feature of a phase portrait.
Figure 1.16 Phase portrait of the flow required for Example 1.4.2. The points P,,., are fixed points. The open sets A, B have boundaries dA, dB, respectively. C is the complement of the closure of A u B .
20 1 Diffeomorphisms and flows 1.5 Conjugacy
Definition 1.4.3 A limit cycle is a closed orbit y strch that either y s L,(x) or y s L,(x) for some x 4 y .
Theorem 1.4.1 has the important corollary that a nonempty, compact set A which is positively or negatively invariant contains either a limit cycle or a fixed point. This result can be useful in demonstrating the existence of limit cycles (Arrowsmith & Place, 1982, pp. 14751).
15 Conjugacy
We now turn to the equivalence relations which allow us to recognise when two diffeomorphisms or two flows exhibit the 'same' behaviour. These equivalence relations lie at the heart of topological or qualitative theory.
Definition 1.5.1 Two [email protected] f, g: M + M are said to be topologically (or CO) conjugate i f there is a homeomorphism, h: M + M, such that
h.f =g.h. (1.5.1)
Topological conjugacy of two flows Q,, $,: M + M is defined in the same way with (1.5.1) replaced by hcp, = +,.h for all t~ W.
Definition 1.5.1 means that h takes each orbit off (or Q,) onto an orbit of g ($,) preserving the parameter m (t), i.e.
P ( x ) 3 gm(h(x)), for each r n ~ Z, (1 S.2)
Q,(x) 3 #,(h(x)), for each t E R. (1.5.3)
The significance of (1.5.2 and 1.5.3) is illustrated in Figure 1.17. Notice, by uniqueness of the trajectories of each flow, a given trajectory of Q, is mapped onto one and only one trajectory of $, and vice versa.
Example 1.5.1 Let f: R + R be a diffeomorphism with Df(x) > 0 for some x E 88. Given that the differential equation i = f(x)  x defines a flow cp,: R t R, show that f is topologically conjugate to cp,.
Solution. Iff is a diffeomorphism it is either an increasing or a decreasing function (differentiability of f' means that Df can never become zero). Since, Df(x) > 0 for some x, it follows Df(x) > 0 for all x and f is an increasing function (see Figure 1.18). It follows that f can have any number of fixed points (including zero). Such points, x:, i = 1,2, . . ., are given by xf = f(x:) and clearly coincide with the singular points of the vector field J(x)  x.
Let xo be any point of the open interval ( x t , xf+ ,). The orbit of x, under both
Figure 1.17 Diagram illustrating conjugacy of: (a) dilfeomorphisms; (b ) flows. Note that (b) is valid for all IER and (1.5.1) implies that h(f"(x)) = gm(h(x)) for all m E E .
Figure 1.18 Typical graph of a dilfeomorphism f: R + LB for which Df(x)>O for some XER. The fixed points of f are given by the intersections of the curve y = f(x) and the straight line y = x.
22 I Diffeornorphisms and flows 1.5 Conjugacy 23
f and cp, is confined to this interval and has the same orientation for both maps (N.B. sign(x,, ,  x,) = sign(f(x,)  x,) = sign(i), n E Z).
Let x,, yo E (x:, xi*, ,) and consider the orbit of x, under f and the orbit of yo under cp,. Let P, = fn(xo) and Q, = cp,(yo), n E Z. Observe (see Figure 1.19) that
J:CPn,Pn+ll+CPn+1,Pnt21
and (1 5 4 )
c ~ I : C Q n ~ Q n + l I  + C Q n + l * Q n + 2 1 ,
n e Z, are orderpreserving diffeomorphisms. Moreover, if x E [Po, P, J then fn(x)e[P,, P,,, ,] and similarly with x, P and f replaced by y, Q, cp,.
Our aim is to construct a homeomorphism on [x:, x:, ,I taking orbits of cp, onto orbits off, preserving the parameter neZ. To this end, let h,: [Q,, Q,] + [Po, PI] be a homeomorphism, for example we might take
Now, for Y E [Q,, Q,, ,I , define
Clearly, h,: [Q,, Q,, ,I + [P,, P,, ,] and, what is more,
Figure 1.19 Orbits of the points x, and yo under f and cp,, respectively. It is convenient to define P, = f "(x,) and Q, = cp,,(yo), for ne Z.
It follows that h: [x:, xi*+ ,] + [x:, xi*, ,] defined by
is a homeomorphism. Finally, it is easy to verify that h exhibits the conjugacy of f and cp,. If
x E [x:, x:, ,] then x E [Q,, Q,, ,] for some n and
as required.
It is important to note that Example 1.5.1 highlights a special property of some increasing diffeomorphisms of the line. Not all diffeomorphisms on R are topologically conjugate to the timeone map of some flow. For example, if j is a decreasing diffeomorphism on W, the orbits off oscillate about its fixed point (see Figure 1.20). Such behaviour is impossible for the timeone map of any flow on R.
If h, in Definition 1.5.1, is a Chdiffeomorphism with k > 1, rather than a homeomorphism then I and g (or Q, and $,) are said to be Ckconjugate. This kind of conjugacy is far more restrictive than topological conjugacy. For example, the real valued functions f(x) = 2x and g(x) = 8x, x E R, are topologically conjugate but they are not Ckconjugate for any k 2 1 (see Exercise 1.5.1). Ckwnjugacy of
Figure 1.20 Graphical derivation of a typical orbit of a decreasing diffeomorphism f: R , R in the neighbourhood of its &xed point. The orbit clearly oscillates from one side of the fixed point to the other.
I /
24 I Diffeomorphisms and Jows 1.5 Conjugacy
cp, and $,corresponds to there being a ktimes differentiable change of coordinates, h, which transforms the differential equation, x = X(x), of cp, into that, y = Y(y) say, of $,. Recall 6conjugacy of cp, and $, means that there is a function h e Ck such that h(cp,(x))=$,(h(x)). Differentiate this equation with respect to t and evaluate at t = 0, to obtain
since cpo=id,. Now consider the change of coordinates y = h(x) applied to x = X(x). With the aid of (1.5.1 I), we find
as required. Thus, when h exhibits the conjugacy of ~p and $, the derivntioe map, Dh, transforms the vector field X(x) into Y(y) with y = h(x).
An important example of C'conjugacy of flows occurs in the qualitative study of local phase portraits in the neighbourhood of an ordinary point. Let x, be an ordinary point of the flow cp: W x W" + Rn of the vector field X: Rn + 88".
Definition 15.2 A local (cross) section at x, is an open set, S , containing xo, in a hyperplane H G Wn which is transverse to X(x,).
For convenience, we will assume that H has normal X(x,) in the following discussion. Observe, (see Figure 1.21) that there is a neighbourhood, V, of x, such that any point X E V can be written as x = cp,(y), where y E S . In other words, we can use the trajectories of the flow to define new coordinates on V.
These new coordinates are best related to local coordinates at x,, therefore, let X H X  x,, SO that xo is at the origin of both sets of coordinates. Now suppose we choose a basis in Rn which has X(0) as its first vector. Then the first coordinate of every point y E S is zero and S defines a neighbourhood, ?, of the origin in Wn (see Figure 1.21(b)). Each point of S can be specified by C E Rn' and every point x of V can be written as
By definition of 9, h: R" + Rn is a C1function. What is more, hlS" is the identity and D,h(O) = X(O), by (1.3.3). Thus Det Dh(0) # 0 so that hI exists and is C1 by the Inverse Function Theorem. In the new coordinates, the trajectories of the flow are simply lines of constant e (see Figure 1.21 (c)), i.e.
Figure 1.21 Various representations of the 'flowbox' containing the ordinary point x,: (a) in the original coordinates; (b) in local coordinates at x, and (c) using local coordinates defined by the flow lines.
26 I Diffeomorphisms and Bows 1.5 Conjugacy 27
To show that el and Q, are conjugate, observe that (1.5.14) implies
However, (1 S.13) gives
by (1.3.2). Thus h(k(4 t)) = Q,(h(u, t)) (1.5.17)
and $, is C1conjugate to Q,. The arguments presented above essent.ially constitute a proof of the 'Flowbox' Theorem.
Theorem 1.5.1 (Flowbox) Let x, be an ordinary point of the flow Q. Then in every sujiciently small neighbourhood of xo, Q is C1conjugate to the flow +(t, x) = x + te,, where el is a unit vector parallel to the xlaxis.
The above examples emphasise that in order to prove two flows or diffeomorphisms conjugate, we must construct an appropriate map satisfying (1.5.1). It is often a great deal easier to recognise when no such map exists. For example, consider two flows: Q, with an isolated fixed point and $, with no fixed points at all. The fixed point is a trajectory of Q, and, therefore, if Q, and (CI, are topologically conjugate, there is a homeomorphism which takes a trajectory of +, onto the fixed point. However, every trajectory of 9, contains more than one point and can only have a single point image under a noninjective map. This contradiction proves that Q, and $, are not topologically conjugate. This result has an obvious extension: a necessary condition for two flows to be Coconjugate is that they have the same number of fixed points. Here an easily recognisable property of the flows (namely, the number of fixed points) allows us to conclude that they are not conjugate.
Another, perhaps less trivial example of this approach, is afiorded by dilfeo morphisms on the circle. Let us begin by considering pure rotations.
A property that distinguishes rational and irrational rotations is their rotation number. This quantity can be defined for any homeomorphism f: SL + S1.
Definition 1.53 The rotation number, p( f ), of a homeomorphism f: S1 + S' is given by
 In(.). modl , p( f ) = Lim Lrn n )
where .T is a lift off.
As Figure 1.22 shows
is a lift of the pure rotation Ry(0) = (0 + y) mod 1. Thus &(x) = x + ny and p(R,)= y, i.e. the rotation number of R, is simply y itself. A rational rotation, R,, a = p / q ~ Q , cannot be topologically conjugate to an irrational rotation, Rp, PER\Q. We saw in 9 1.2.2 that the orbit of any point 0 under R, was periodic with period q, i.e. RZ(8) = 0, while R;(O) # 0 for any B E [O,2n) or ~ E E . Clearly, any map taking an orbit of Rp onto an orbit of R, would fail to be injective. Therefore, the pure rotations with rational rotation number are topologically distinct from (i.e. not Coconjugate to) those with irrational rotation number. Now, the pure rotations are diffeomorphisms on S1 and p( f ) is defined for any diffeomorphism f: S1 + S'. To what extent, therefore, can the above result for pure rotations be carried over to general diffeomorphisms on S'?
Proposition 15.1 A dfleomorphism f: S1 + S' has periodic points if and only if its rotation number, p( f ), is rational.
Proof. Iff has a periodic point then, given a lift,x of f, there exists x* E R such that
T(X*) = X* + p, (1.5.20)
for some integers p and q. It follows that p ( x * ) = x* + np, and therefore
Hence p( f ) is rational. To prove the converse, suppose f has no periodic points then,
for any integers p, q and any X E W. Sinceg,(x) = f4(x)  x satisfies g,(x + 1) = g,(x),
Figure 1.22 Commutative diagram illustrating the connection between the pure rotation R, and its lift R,.
As our notation suggests, it can be shown that p( f ) is independent of the point x occurring in (1.5.18). A proof of this fact can be found in Nitecki (1971, pp. 334). = (X + a) mod 1
28 I Diffeomorphisms and jlows 1.6 Equivalence of jlows 29
for each x, (1.5.23) means that there exists E > 0 such that either
g,(x) < p  E, for all x; (1.5.24)
g,(x) > p + E, for all x. (1.5.25)
Suppose (1 3.24) holds, then f4(x) < x + p  E, for all x, and therefore
Similarly, when (1.5.25) is valid
J"'(x) > x + n(p + E). (1 5 2 7 )
Thus, ~im[j"4(x)  x]/nq is either greater than (p + ~ ) / q or less than ( p  &)/q, n4m
for any integers p and q, and so p( f ) # (plq) mod 1.
Typically, circle diffeomorphisms with rational rotation number, p / q ~ Q, have an even number of periodq cycles. A sketch of p ( x ) (see Figure 1.23) not only reveals why the number of cycles is even, but it also shows that the stable and unstable points alternate around the circle.
The following result shows that circle diffeomorphisms with irrational rotation number can behave like irrational rotations.
Theorem 1.5.2 (Denjoy) If an orientationpreserving d$eomorphism f: S' + S' is of class C2 and p( f ) = PE R\Q, then it is topologically conjugate to the pure rotation
RP
For a proof of Denjoy's Theorem the interested reader should consult Arnold (1983, pp. 1056) or Nitecki (1971, pp. 459). This important result means that every orbit of f is dense in the circle provided f EC' and p( f ) is irrational. If f$C2, then more complicated phenomena, such as invariant Cantor sets, can occur (see 9 6.4.1 and Nitecki, 1971).
1.6 Equivalence of flows
Topological conjugacy is arguably the natural equivalence relation for maps. A homeomorphism h is used to take successive points in the orbit of one map, f, onto those of another map, g. Given that the aim is to capture the fact that the orbits o f f and g bebave in a similar way, continuity of h and its inverse is the least we should demand. Moreover, since the orbits of a map are sequences of discrete points, it is hard to envisage anything more sensible than mapping orbits onto orbits in the manner described above. However, this is not the case for flows.
From this point of view, the important difference between maps and flows is that
the orbits of the latter are parametrised by a continuous variable t. This allows us some additional freedom in the mapping of orbits onto orbits.
Definition 1.6.1 Two flows, Q, and $,, are said to be topologically (or CO) equivalent $ there is a homeomorphism, h, taking orbits of cp, onto those of $,, preserving their orientation.
Since equivalence only demands that orientation be preserved, we allow h(~,(x)) =
$r,a(~), with y = h(x), where r , is an increasing function oft for every y (see Figure 1.24). This relaxation of the requirement that the parameter t be preserved, provides more satisfactory equivalence classes for flows. For example, the planar differential
Figure 1.23 (a) Sketch of p(x). Observe that, since f'(1) =f'(0) + 1, if a fixed point, x,*, occurs then there must be at least one further fixed point x:. Moreover, if xg is stable then x: must be unstable. (b) Example of J 3 ( x ) for a circle diffeomorphism with a stable 3cycle. Note that an unstable 3cycle must also occur. (c) Illustration of periodic points off on the circle for the lift shown in (b) .
30 I Diffeornorphisms and flows 1.6 Equivalence of flows 31
equations
1 r ) , 6 = 1, (1.6.1)
i = r(1  r), 6 = 2 , (1.6.2)
where (r, 6) are polar coordinates, have similar phase portraits. Both have an attractive closed orbit y with r ( t )  1 and an unstable focus at the origin. However, the closed orbit has period2n in (1.6.1) and periodn in (1.6.2). Thus, if h: y + y preserves the parameter t, it must fail to be a bijection. Thus (1.6.1) and (1.6.2) are not topologically conjugate, but they are topologically equivalent. Observe that the time rescaling t 2t transforms (1.6.2) into (1.6.1).
If Definition 1.6.1 is satisfied with h e c k , k >, I, then the stronger relationship between Q and $ can be emphasised by saying that they are Ckequivalent. If two flows Q and $ are Ckequivalent (k 2 0) then their vectors fields X(x) and Y(y) are also said to be Ckequivalent. This terminology is frequently used because flows are often described implicitly in terms of their vector fields. For example, in applications one is olten provided with a model differential equation but no explicit form for its solutions.
When k >, 1 there is a Ckdiffeomorphism, h, such that
(cf. (1.5.11)), where a : R" r R takes only positive values corresponding to the reparametrisation of the time. Recall the vector field of SZy,,, is given by
Figure 1.24 Topological equivalence requires trajectories to be mapped onto trajectories preserving their orientation rather than t itself. Thus, r , ( t ) is an increasing function oft that is continuously parametrised by y and satisfies r,(O) = 0. For example, when r,(t) takes the form shown in (a), h relates cp,(x) and #,y(,,(h(i)) as indicated in (b).
where a(y)= f,(O) is a positive scale factor altering the magnitude but not the direction of Y(y).
Example 1.6.1 Show that the vector fields J x and J,x, with
where a, /I > 0, are topologically equivalent.
Solution. The differential equations x = J x and x = J,x are easily solved using plane polar coordinates. We find x = J x gives i = ur, 0 = f l with solutions
r(t) = r, exp(at), 6 = ,?t + 8,. (1.6.6)
The equation x = J,x becomes R = R, 6 = 1 and its solutions are
R(t) = R, exp(t), Q = t + 0,. (1.6.7)
If we let t H Pt in (1.6.6), we obtain
r(t) = r, exp(at//?), 0 = t + 0,. (1 6.8)
Since j > 0, the flows defined by (1.6.6) and (1.6.8) are topologically equivalent with h equal to the identity. In other words, they have identical trajectories and differ only in the speed at which they are described.
Elimination of t from (1.6.7) and (1.6.8) gives
Equation (1.6.9) defines a map taking the trajectory of (1.6.8) through (r,, 6,) onto the trajectory of (1.6.7) through (R, , 0,) (see Figure 1.25). For r, r, > 0, this map is 1:1, continuous and preserves orientation (indeed it preserves t itself);
Figure 1.25 Illustration of the effect of the map (1.6.9) on the orbit of (1.6.8) through (r,, 0,). The result is the orbit of (1.6.7) passing through ( 4 9 @,I.
32 1 Difleomorphisms and jlows 1.7 Poincare maps and suspensions 33
however, it involves four parameters. In fact, (1.6.9) represents a family of maps of the plane onto itself. We require a single homeomorphism taking each trajectory of (1.6.8) onto an orbit of (1.6.7) and, therefore, we must choose values for the parameters.
Observe, every trajectory of (1.6.7) and (1.6.8) crosses the unit circle once and only once. Let us choose to map the orbit of (1.6.8) that crosses the unit circle at angular coordinate 8, onto the orbit of (1.6.7) that crosses the unit circle with angular coordinate O, = 0,. The map h obtained in this way is given by setting ro = R, = 1 and 8, = 0, in (1.6.9), i.e.
R =rfl/a, @ = f ~ , (1.6.10)
with r >O, 0 6 0 < 2n. Thus, if we define h(O)= 0, we have constructed a homeomorphism which exhibits the topological equivalence of (1.6.7) and (1.6.8). Since we have already established the equivalence of (1.6.6) and (1.6.8), we finally conclude that J x and Jox are topologically equivalent.
Example 1.6.2 Use the map r' = r, 8' = 9 In r (r > 0) to demonstrate that the vector fields Jox, where J, is given in (1.6.5), and x are topologically equivalent.
Solution. Let h be given by
h(x) = h(r cos 0, r sin 8)
  (;:In r), r sin(9  In r), r > 0 (1.6.11)
r=O.
The map h: R2 + R2 is continuous and has continuous inverse, r = r', fJ = 0' + In r', r' > 0. Since h(0) = 0, h takes the fixed point trajectory of the flow of Jox onto that of x. For x # O , h is differentiable so we can check its effect on the flow by transforming the differential equation x = Jox or, in polar coordinates, i = r, fJ = 1. We find
. . + . f ' = i = r = r ' and @ = f ~   = e  l =o, (1.6.12)
r
which is just the polar form of x = x. Of course, h is not differentiable at the origin so that (1.6.1 1) is only a homeomorphism of the plane. Hence Jox and x are topologically equivalent.
When two flows are topologically equivalent we say they are of the same topological type. The results obtained in Examples 1.6.1 and 1.6.2 play an important role in the classification, up io topological type, of all linear vector fields on 88' (see Arrowsmith & Place, 1982, p. 58). The matrix J in (1.6.5) is the real Jordan form of any 2 x 2 real matrix, A, with complex eigenvalues a f ip, a > 0, i.e. there is a real nonsingular matrix M such that M'AM = J. It follows (see Exercise 1.5.6)
that the flows of Ax and Jx are linearly conjugate. Examples 1.6.1 and 1.6.2 show that all such vector fields are topologically equivalent to the vector field x.
The complete classification of linear vector fields on Bg2 is summarised in Figure 1.26. Each point of the (Tr A, Det A)plane represents a similarity class of real, 2 x 2 matrices. The striking feature is that the vast majority of points in Figure 1.26 correspond to vector fields of stable, unstable or saddle type. Such linear vector fields are said to be hyperbolic (see $2.1) and Figure 1.26 suggests that hyperbolic behaviour is 'typical' for linear vector fields on R2. The point to note is that, without a suitable equivalence relation, the idea of what is typical has no meaning. We will return to the question of typical or generic properties of flows and diffeomorphisms in $ 3.1.
1.7 Poincark maps and suspensions
We have already noted that the flow map 9,: M * M is a diffeomorphism for each fixed t. Thus, one way of obtaining a diffeomorphism from a flow is to take its timeT map, 9,: M + M, T > 0. Clearly, the orbits of cp, are constrained to follow the trajectories of the flow because {cpT(x)lm E E} = {cp,,(x)lm~ E} E {cp,(x)(t~ W). This means that the dynamics of cp, are strongly influenced by the flow Q and they are not typical of those ofdiffeomorphisms on M. It is perhaps worth stressing that, while the orbits of x under cp,, and cp,,, T, # r,, behave in a similar way for any X E M (because both are subsets of the same trajectory of cp), the two maps are not necessarily of the same topological type. For example, suppose cp, has a closed orbit y of period T and that r , = aT, a € Q, whilst T, = PT, P E R\Q. It follows that p,, has an invariant circle y consisting entirely of periodic points (of period q if a = plq). The same closed curve y is invariant for cpt2 but the orbit of any point x ~ y under cp,, fills out y densely. Therefore, cp,, cannot be topologically conjugate
Figure 1.26 Topological types of all linear vector fields on the plane. Each point in the (Tr A, Det A)plane corresponds to an equivalence class of linear vector fields. Details of the derivation of this diagram are given in Chapter 2 of Arrowsmith & Place (1982). The differential equation x = x has Tr A = 2, Det A = 1 and is therefore unstable.
Det A I
STABLE ?, UNSTABLE g
 non  simple
34 I Diffeomorphisms and flows 1.7 Poincari maps and suspensions 35
to g,,, i.e. the maps are of different topological type. We will have cause to return to timeT maps of flows in Chapter 5.
Another, more significant, way of obtaining a diffeomorphism from a flow is to construct its Poincari map. Let g be a flow on M with vector field X and suppose that Z is a codimension one submanifold of M satisfying:
(i) every orbit of g meets C for arbitrarily large positive and negative times;
(ii) if X E C then X(x) is not tangent to C.
Then C is said to be a global (cross) section of the flow. Let y € C and s(y) be the least, positive time for which g,,,(y)~C.
Definition 1.7.1 The Poincark (or Jrst return) map for Z is defined to be
P(Y) = CP,,)(Y), YE z (1.7.1)
Example 1.7.1 Obtain the Poincark map, P, of the flow defined by
1 r ) ; O = 1, r>O, (1.7.2)
where (r, 8) are plane polar coordinates, taking Z to be the halfline 8 = 0. How does P change if C is taken to be the halfline 0 = 0,?
Solution. The phase portrait of (1.7.2) is shown in Figure 1.15. C is the positive xaxis in the plane and (1.7.1) can be written
p(x) = (~,)(x, o)),, x > 0, (1.7.3)
where q,(r, 0 ) ~ R2 is the flow of (1.7.2), (.), denotes the xcomponent of ., and r(x) is the time taken for a phase point at X E Z to make one complete revolution about the origin. Since 8 = 1, r(x) = 2n.
The radial equation, i = r(l  r), has solution
with r(0) = r,, so that
where a = exp( 2n) < 1. If C is taken to be the halfline 0 = 8,, then (1.7.3) is replaced by
where t(r) = 2n and (.),denotes the radial component of  . We, therefore, conclude that P takes the form (!.7.5) with x replaced by r, the radial distance along 0 = 8,.
of flows in one higher dimension. For example, the Poincark map P(x) in (1.7.5) has a fixed point at x = 1 (observe x* = P(x*) implies (1  x*)(l  a) = 0, which is only satisfied for x* = 1). Furthermore, if x >( 1, then P(x) 5 x so that x = 1 is an attracting fixed point. This fixed point in P clearly corresponds to the stable limit cycle in the phase portrait of g (see Figure 1.15).
Another example is afforded by the flow on the torus, T2, defined by
where 0 and rp are as shown in Figure 1.27. The equations (1.7.7) have solutions
0 = a t + O 0 and rp=flt+cp, (1.7.8)
/It, = 2n reduced mod 271, so first returns to when ', where
0 I:::. . Thus if atg = 2n
alp = plq, p, q e H t and relatively prime, then qt, = pt, and the orbit through (0,,rp0) returns to this point after q revolutions around the torus in the rpsense and p revolutions in the 0sense. It follows that if a and /I are rationally related then every point of T 2 is a periodic point of the flow, i.e, every point lies on a closed orbit. If on the other hand a and fl are not rationally related then the orbit through (0,,rpO) never returns to that point although it approaches it arbitrarily closely.
A global section of the torus is obtained by taking cp = rp,, a constant, when C is a circle, S1, with coordinate 8. Since the orbit of the flow first returns to rp = cp, after time t, = 2x18 and 0 = at + do, we conclude that the Poincare map, P: S' , S' , is a rotation by 2nalfl. The properties of pure rotations (see $1.2.2) obviously reflect the behaviour of the flow described above.
There are flows for which there is no global section (see Exercise 1.7.2). Therefore, it is not true to say that every flow corresponds to a diffeomorphism by taking Poincark maps. However, the converse is true, i.e. every diffeomorphism f is the Poincark map of a flow  called the suspension of f . This is a very important
Figure 1.27 Diagram showing how the coordinates, 0 and cp, used in (1.7.7) are defined.
By construction P: C t Cis a diffeomorphism and dim I: = dim M  1. In contrast to time? maps we, therefore, expect these diffeomorphisms to reflect the properties
36 I Dl~eomorphisms and flows 1.7 Poincare maps and suspensions 37
observation. It means that any result that can be proved for diffeomorphisms should have a counterpart for flows in one higher dimension (see Smale, 1967). The following explicit definition is given on p. 59 of Arnold & Avez, 1968.
Definition 1.7.2 The flow
where X E M , OECO, I] and [.I denotes the integer part of ., defined on a compact maniJold by identification of (x, 1) and (f(x),O) in the topological product M x [0, I], is called the suspension of the difleomorphism f: M t M.
It is easy to verify that #,(x, 8) in (1.7.9) formally satisfies the requirements of Definition 1.3.1. Geon~etrically, (1.7.9) corresponds to considering the product M x [O,1] and taking a unit vector field in the [0, I]direction. Now imagine identifying the Iend and the 0end in such a way that (x, 1) is attached to (f(x), 0) for each x E M (see Figure 1.28).
It must be pointed out that the manifold 2 is not always M x S' as Figure 1.28 suggests. M x S1 is obtained if f is continuously deformable, through diffeomorphisms, to the identity. For example, if we let M = S1 and f be a rotation then fi is the torus T2 = S f x SL. However, if f is a reflection in a diameter of the circle then 2 must be a Klein bottle to achieve the identification of (x, 1) and (f(x), 0). Another, perhaps simpler example is to let M = (0, 1) and f be reflection in x = f. As Figure 1.29 shows, fi is a Miibius strip.
Figure 1.28 Schematic illustration of the construction of the suspension of a diNeomorphism f that is continuously deformable into the identity: (a) before; (b) after; identification of (x, 1) and (f(x), 0).
An alternative way of viewing Definition 1.7.2 is to think of linking (x, 1) and (f(x), 0) by a smooth 'fibre' of unit length along which the suspension is considered to flow. This must be done for each X E M. Since f is a diffeomorphism, if y E M is close to x then f(y) is close to f(x) and the fibres of the identification lie close to each other. If we were to take a finite sample of these fibres we should obtain something resembling unit length of a, possibly twisted, multicored electrical flex.
Obviously, this procedure does not define the precise shape of the identifying fibre or, in other words, it does not uniquely determine the suspended flow. What is important is that the component of the flow in the new dimension is never zero. It then follows that all admissible shapes of the identifying fibres give rise to topologically equivalent suspended flows. The flow given in (1.7.9) is a particular
Figure 1.29 The suspension of the diNeomorphism f: ( 0 , l ) , (0, 1) given by reflection in x =$ is defined on a Mobius band. The twist in the manifold on which the suspension is defined arises because (x, I ) must be identified with ( f ( x ) , 0).
38 1 Diffeomorphisms and flows 1.8 Periodic nonautonomous systems
representative of this equivalence class which clearly exhibits the connection with the dilfeomorphism f. When looked at from this point of view it is easier to understand how the nature of f (whether or not it is deformable to id,) affects the resultant manifold on which the suspension is defined.
1.8 Periodic nonautonomous systems
An important application of the ideas developed in $1.7 is in the analysis of differential equations of the form
i=X(x , t ) , X E M , (1.8.1)
where
X(x, t + T) = X(x, t), (1 3.2)
for all t~ W. The transformation t wtly, X(x, t ) ~ yX(x, yt), with y = T/2n, allows (1.8.1) to be written as the autonomous system
ic = X(x, O), 6 = 1, (1 3.3)
defined on M x BB, where
X(x, O + 2n) = X(x, 8)
for all O E W (see Exercise 1.8.1). It is then convenient to identify O + 2nm, m EZ, with 8 to obtain a differential equation on M x S1, where 8 is the circular coordinate. This procedure is illustrated in Figure 1.30 where some possible solutions of (1 3.1 and 2) are shown. Observe that the solutions are not necessarily periodic (see Exercise 1.8.2). However, it is easily verified that if C(t) is a solution of (1.8.1 and 2) then so is { ( t + T) (see Exercise 1.8.2). i.e. advancing a solution by one period of the vector field also gives a solution.
Figure 1.30 helps us to associate this 'period advance map' of the non autonomous system with the Poincark map P,: C, , C, of (1.8.3) defined on the global section, Z, = M x {O) of M x S1. It is worth noting that P, and P,., 8 # 8' are topologically conjugate (see Exercise 1.8.4). Thus, in discussing topological properties it is suficient to consider P= Po. Conversely, we can associate the solutions of the nonautonomous system with the suspension, on M x S1, of the Poincark map, P, which is itself a diffeomorphism on Co = M x 10).
There is a complete correspondence between the properties of the Poincark map, P, and those of its suspension. For example, P has a f ixed point x* if and only if its suspension has a closed orbit o j period 271, i.e. if and only if the nonautonomous system has a periodiq solution of period T. Figure 1.30 shows a 2cycle of P along with the corresponding solution of (1.8.1,2) with period 2T. Furthermore, a periodic solution of (1.8.1,2) is stable (asymptotically stable), in the sense of Liapunov, if and only if the associated periodic point of P is stable (asymptotically stable). The following example shows how this last result can be applied.
Figure 1.30 (a) Schematic representation, in the extended phase space, M x R, ofsome possible solutions of the nonautonomous system (1.8.1), (1.8.2). ( b ) Corresponding solutions of the autonomous equation (1.8.3), (1.8.4)on M x S'. C,= M x 10) is a global section for the flow of(1.8.3), (1.8.4) and this allows us to deline the Poincare map Po: Co + Z,.
2x1 0 =  T
mod 2n
&(I) s 0
period 4rr
period2x x ; = P j x ;
x * = P O x * Zg = M x { ~ ]
(b)
40 1 Dif/eomorphisms and flows 1.8 Periodic nonautonomous systems 4 1
Example 1.8.1 Find the period advance map for the nonautonomous system
where w(t) = w(t + T), t~ R. Obtain the Poincark map P and show that Det P = 1. Hence, deduce that the null solution of (1.8.5) is stable (in the sense of Liapunov) if JTr PI < 2 and unstable if ITr PI > 2.
Solution. The secondorderequation (1.8.5)can be written in thelirstorder form
x = A(t)x, (1 3.6)
where x = (s,, x , ) ~ = (x, . i ) T ~ RZ and
The solutions of (1.8.6) form a twodimensional vector space (see Exercise 1.8.3). The solution r(t) satisfying {(to)=xo can be written in the form
where the columns of Q(t) form a basis for the solution space of (1.8.6). Q(t) is called a ,fundamental matrix for the problem (see Jordan & Smith, 1977) while q(t, to) is known as the state transition matrix (see Barnett, 1975). Now observe that
{(t + T ) = Q(t + T)Q '(to)x0 = Q(t + T)Q '(t)Q(t)Q '(t0)x0
= cp(t + T, t){(t).
Thus cp(t + T, t): C, +C,+T (see notation in Figure 1.30) is the period advance map at t. Clearly, if { and q are solutions of (1.8.6) then
cp(t + 7; t)(a€(t) + W ) ) =acp(t + T, t)t(t) + bp(t + T, t)q(t),
a, h < R, and the period advance map is linear for any t. Moreover,
cp(t+T,t)=Q(t+T)Q'(t)
= cp(t + T, T)cp(T, O)cp(O, t). It can be shown that: ( i ) cp(t + T, to + T) = p(t, to); (ii) cp(t, 0) ' = q(0, t);
(see Exercise 1.8.4) so that (1.8.9) can be written in the form
This result shows th& cp(t + T, t) and cp(T, 0) are topologically (indeed linearly) conjugate and consequently, for the qualitative behaviour of the solutions of (1.8.6), we can focus attention on q(T, 0). Expressed in terms of 0, y(T, 0) = Po =
P: R2 + RZ, the Poincark map for (1.8.6).
To show that Det P = 1, note that (1.8.8) implies that
since ~ ( t ) = A(t)Q(t). It follows (see Exercise 1.8.6) that, if W(t) = Det(cp(t, 0)), then ~ ( t ) = Tr(A(t))W(t) = 0 for (1.8.6). Hence W(t) = W(0) = Det(p(0,O)) = 1 and, in particular,
W(T) = Det(q(T, 0)) = Det P = 1. (I .8.12)
The null solution of (1 3.6) corresponds to the fixed point of P at the origin. The stability type of the null solution is the same as that of the fixed point and the latter is determined by the eigenvalues, I.,,,, of P. Since Det P = 1, the characteristic equation of P is 1'  (Tr P)R + 1 = 0 and
A,,, = $(Tr P _+ [(Tr P)'  4]'12). (1.8.13)
If ]Tr PI < 2 then (Tr P)2 < 4 and the eigenvalues are complex with I, = 1; = exp(i/?) (since R 1 l , = I), where tan /?=[4 (Tr P)Z]'lZ/Tr P. Let u +iv, u, v€BB2, be the eigenvector of P with eigenvalue A,. Then the matrix K = (viu) is such that
cos f l sin f l K'PK=
sin /? cos f l i.e. P is conjugate to a rotation about x =O. It follows that the orbit of x f 0 under P lies on an ellipse and, consequently, the fixed point at x = O is stable in the sense of Liapunov (see Figure 1.31).
If ITr PI > 2, then L,., are real with A, = I . ((I1 > 1) 1, = I'. In this case, there is a nonsingular K such that
Figure 1.31 When ITr PI < 2, the orbits of points x # O under P lie on ellipses as shown. Observe that, for any XEN', P x e N for all m€Z. Thus, x = 0 is stable in the sense of Liapunov (see Definition 1.2.3).
42 1 DifSeomorphisms and flows 1.9 Hamiltonian flows and Poincare maps 43
Here the orbits of P lie on hyperbolae and, as Figure 1.32 shows, the x = 0 is an unstable fixed point.
Example 1.8.1 suggests that periodic perturbations of the frequency, w, of a harmonic oscillator can destabilise the equilibrium point with x = O . This is essentially what a child on a swing achieves by appropriate movements of weight, in order to build up the amplitude of the oscillations of the swing. A simple example illustrating how tl~is instability can be achieved is given in Arnold (1973, pp. 2054). This phenomenon is known as parametric resonance.
1.9 Hamiltonian flows and PoincarC maps
Another application of Poincark maps, that is of current research interest, lies in the study of nonintegrable, conservative Hamiltonian systems. While the reader will no doubt have encountered the integrable case in a Classical Mechanics course, it will be useful to review the basic ideas emphasising the connection with flows.
Definition 1.9.1 Let U be an open subset of R2" and H: U + R be a twice continuously differentiable function. The systetn of drflerential equations x = X,(x), X,,: U + 88'" given by
where x = (q , , . . ., q,, p,, . . ., P , ) ~ is said to be a conservative Hamiltonian system with ndegrees of freedom.
Figure 1.32 For i. > 1 the hyperbolae x,x, = c, c # 0, are invariant curves for the map Dx, where D is given by (1.8.15). The origin is a hyperbolic saddle point and therefore for every N ' c N, there exists X E N o for which Pmx$ N , for some rnE Z+. Hence the saddle point is unstable in the sense of Liapunov.
H = H(q, p) is the Hamiltonian for the system and the equations (1.9.1) are known as Hamilton's equations. The state of the system at time t is specified by
The conjguration, q(t), of the system is given by the n generalised coordinates q,(t) and p(t) consists of the n conjugate generalised momenta pi(t). A system with n degrees of freedom is often called an nF system.
In general, q, and pi change with t but H does not. Observe
for all t, by (1.9.1). Thus, H is a conserved quantity or a constant of the motion. Alternatively, (1.9.1) is an autonomous system of differential equations which defines a Hamiltonian flow, qf: U , R2". Equation (1.9.2) means that H is constant on the trajectories of cpp i.e. H is a first integral for (1.9.1) (see Arrowsmith & Place, 1982, pp. 1016).
In general, Hamiltonian flows occur on differentiable manifolds and Definition 1.9.1 is valid for each chart (U,, ha). Thus (see Figure 1.33) Ha: U, + R gives rise to a vector field X,., via (1.9.1), for each a. Moreover, when Wan Wg (a # P ) is nonempty, the two sets of local coordinates on U, and Ug are related by the
Figure 1.33 Illustration of the way in which a Hamiltonian function defined on a manifold M gives rise to Hamiltonians, Ha and Hp on the charts (U,, ha) and (Uc , hb). respectively.
44 I Diffeomorphisms and flows 1.9 Hamiltonian flows and Poincare maps 45
overlap map hao (see Figure 1.3). Thus, if x = (q , , . . ., q,, P I , . . ., pJT in Ua and Y = ( Q ~ , . . ., en, P , , . . ., in UB represent the same point on M, then
H,(x) = HB(haB(x))9 (1.9.3)
and
H,(~,B' (Y)) = HB(Y 1. (1.9.4)
Of course, we require that the vector fields XHa and XHo give rise to the same dynamics on the overlap between two charts and this imposes constraints on the manifold itself. To make the dynamics on W, and Wg agree on W a n Wg, we demand that
Dxhag(x)X~.(x) = X"p(haB(x)) (1.9.5)
(see (1.5.12)). Now, differentiation of (1.9.3) gives
Equation (1.9.6) looks more familiar in component form, i.e.
Furthermore, (1.9.1) implies
CDxH,(x)lT = SX,.(x)
and
CD,H, (Y) I~ = SXH,(Y),
0 1 with S = (I O) and I equal to the n x n unit matrix. Operating from the left
with [DXhaB(x)lTS in (1.9.5) gives
CDxhap(x)IT~Dxhap(x)XH.(~) = CDXhaa(~)ITSX~p(haB(~))9 (1.9.10)
= [DxhaB(x)ITIDyHp(h~s(x))lT = [DxHa(x)IT, (1.9'1')
by (1.9.9) and (1.9.6), respectively. Finally,
Clearly, (1.9.5) is satisfied if and only if the overlap map hap is such that
CDxh~(x) lTsD,hap(x) = S, (1.9.13)
for each x E ha(Wan WB).
Definition 1.9.2 A dgeomorphisnt h: U + Rzn, U E R2", is said to be symplectic i f
[Dh(x)ITSDh(x) = S (1.9.14)
for all x E RZn, with S = ( :) where I is the n x n identity matrix.
A differentiable manifold for which all the overlap maps satisfy (1.9.13) is said to be a symplectic manifold. The theory of symplectic manifolds provides a coordinate free approach to Hamiltonian mechanics (Abraham & Marsden, 1978; Arnold, 1968).
It is important to realise that (1.9.13) is sufficient to ensure that the form (1.9.1) of Hamilton's equations is valid on both U , and UB (see (1.9.8,9)). The arguments involved in obtaining (1.9.13) are not confined to overlap maps. Consider the effect of a coordinate transformation, h, on a Hamiltonian system defined on R2*. If we demand that the equations of motions of the new coordinates be derived from the transformed Hamiltonian by applying (1.9.1), then we can conclude, by precisely the same steps as we have used above, that h must be symplectic. However, preservation of Hamilton's equations in this sense is the property that defines canonical transformations in Classical Mechanics. Thus symplectic and canonical transformations are one and the same thing.
A property that distinguishes a Hamiltonian flow, (p:, from other flows of even dimension is that cp;H preserves volumes of phase space.
Theorem 1.9.1 (Liouville) Let (p, be the flow induced by x = X ( x ) and Q(t ) be the volume of the image, (p,(D), of any region D of its phase space. If div X r 0 , then (p, preserves volume, i.e. Q( t ) = Q(0) for all t .
To illustrate the ideas behind the proof of Theorem 1.9.1 we will assume that D and (p,(D) both lie in the same chart. Since cp, is a diffeomorphism for each t , we can regard it as a change of coordinates in phase space. With notation in Figure 1.34,
Since x' = (p,(x), this can be written as
46 I Diffeomorphisms and flows 1.9 Hamiltonian flows and Poincare maps 47
where d2"x = dq,, . . . , dp,. Now,
and therefore
Thus
since Q(0) = jD dq,, . . ., dp,. However, observe that if DX(x) has eigenvalues qi(x) then
Det(Dq,(x)) = Det(1t tDX(x) + 0(t2)),
Of course,
Tr DX(x) = div X(x)
and substitution in (1.9.19) gives
h(0) = div ~ ( x ) d ~ " x . ID Figure 1.34 The flow map Q, takes D at time zero (see (a)) to cp,(D) at time t (see (b)). Since Q, is a diffeomorphism, this transformation can be regarded as a change of coordinates from (q , , . . . , q,, p , , . . ., p,) = xT to (q; , . . ., q:, p;, . . .. P:) = xIT.
The above arguments do not depend on the initial time being zero and (1.9.22) can be generalised to
h(t) = div X(x)d2"x. I,,, (1.9.23)
Clearly, if div X(x) r 0 then b(t) r 0 and Q(t) = R(0) for all t .
Let us apply Theorem 1.9.1 to a Hamiltonian flow cpf. The vector field X is given by (1.9.1) and
Hence Q; preserves phase space volumes. This result highlights, in a geometrical way, the very special nature of Hamiltonian flows. In general, even dimensional flows may expand volumes in some parts of phase space and contract them in others. Clearly, (1.9.24) imposes a global restriction on Q:. The volumepreserving nature of Hamiltonian flows is also reflected in the nature of the transformations that relate them to one another. It can be shown (see Arnold, 1968, p. 222 and Exercise 1.9.5) that (1.9.14) implies Det(Dh(x)) r 1 so that symplectic trans formations preserve volumes of phase space. However, it is perhaps worth noting that Det(Dh(x))= 1 only implies h is symplectic when h: W2 + W2 (see Exercises 1.9.5 and 1.9.6).
It is reasonable to consider to what extent Hamilton's equations can be simplified by syrnplectic transformations. Let h: (q, p) + (Q, P)and i ( ~ , P) = H(h'(Q, P)). In particular, the transformed equations will be simpler if the new Hamiltonian is independent of one of the generalised coordinates. For example, suppose t? does not depend on Q,, then
and
The constant value I, can be thought of as a parameter. For a given value of I,, H now depends on only (n  1) pairs of conjugate variables; the number of degrees of freedom has been reduced by one and the order of Hamilton's equations has decreased by two.
Ideally, one would like 17 to be independent of all Qi, i = I,. . ., n. Then
and
48 I Difleomorphisms and .flows 1.9 Hamiltonian flows and Poincari maps
i = I, . . ., n. Notice Qi depends only on the parameters I ,, . . ., I, and is therefore independent of t . Thus (1.9.28) can be trivially integrated to give
i = I , . . ., n, K , E R . Systems for which such a reduction is possible are said to be integrable and the system defined by (1.9.27 and 28) is referred to as their normal form. The variables (Q, P) displaying this form are called actionangle oariuhles; the Pi (or l i ) being the 'actions' and the Qi being the 'angles' (or cyclic variables). The latter name arises because (1.9.27 and 28) is the polar form of a simple harmonic oscillator with radial coordinate Ii and angular coordinate Qi.
Traditional courses in Classical Mechanics focus attention on the integrable case. For example, IF systems with analytic H, linear equations of motion (i.e. normal modes), nonlinear systems that are separable into IF systems are com monly discussed. However, these systems are not typical. In general, Hamiltonian systems are nonintegrable and they can exhibit much more exotic dynamics. TO illustrate this we must consider systems with at least two degrees of freedom and Poincari maps play a key role in making such problems manageable.
A system with twodegrees of freedom has a fourdimensional phase space and it is, therefore, not feasible to picture its flow directly. Since the system is conservative, (generically) its trajectories lie in threedimensional submanifolds or 'shells' on which the Hamiltonian H(q, p) is constant. Thus, by choosing a particular value for H(q, p) we can reduce the dimensionality of the problem by one. Now, we are frequently interested in systems exhibiting some kind of recurrence. For example, nonintegrable perturbations of an integrable system or the behaviour of a nonintegrable system in the neighbourhood of a closed orbit. In such cases, we can reduce our problem to one in two dimensions by constructing an appropriate Poincart map. Of course, we have lost some detail of the dynamics in this process. After all we are only sampling the orbit periodically. However, the interesting point is that sufficient information is retained to show that the dynamics of 2F, conservative systems can be very complicated. Moreover, since this information is in twodimensions it is quite easy to present and appreciate in graphical form.
To show how the Poincart map is constructed, let us first examine an integrable case, where solutions can be written down explicitly. Consider the biharmonic oscillator
The Hamiltonian H(q, p) is given by
and (1.9.30) has solutions of the form
q i € R, i = 1,2. The aim is to construct the Poincark map in such a way that one pair of conjugate variables (q,, p2, say) are removed. Thus we argue that by restricting to the Hamiltonian shell H(q, p) = h, > 0 we can express p, in terms of q,, p , and q2. Since 9 , is periodic with period 2n/w2, the orbit of a phase point in the plane q2 = 0 returns to q , = 0 after time 2n/o, (see Figure 1.35). Therefore, the Poincari map P defined on the section 9, = 0 is given by
1 cos '""sin '31 27cy)(;:)
(1.9.33)  w1 sin 2ny cos 2ny
with y = W~/W,. Clearly, P represents a rotation for which the ellipses
with 0 < C < 2ho, are invariant curves. These closed invariant curves correspond to invariant tori in the flow on the H = h, shell.
The important thing to notice about (1.9.33) is that Det P = 1. This means (see Exercise 1.9.5) that the Poincart map, constructed in the manner described above, is areapreserving. That this is also the case when the system is nonintegrable follows from the PoincarfiCartan invariant (Arnold, 1968, pp. 23340 or Arnold & Avez, 1968, pp. 23Ck2). A derivation of this invariant for 2F systems requires
Figure 1.35 The Poincare map defined on the section q, = 0. It is clear from (1.9.32) that q, returns to zero periodically with period 21r/o,.
50 I Diffeomorphisms and flows I a knowledge of differential forms, however, for IF systems it can be obtained in the familiar notation of vector analysis. Consider the extended phase space for a 1F system with coordinates (q, p, t). Let v = (p, 0, H), then curl v = (dH/dp, dH/dq, 1) is the vector field of the Hamiltonian H in extended phase space (see (1.9.1)). Now apply Stokes Theorem to the tubular region shown in Figure 1.36(a). Here the sides of the tube consist of flow lines of curl v. Dissecting
Figure 1.36 (a) Tubular region to which Stoke's Theorem is applied for IF systems. The vector field curl v is tangent to the surface at every point of the tubeso that curl v.dS E 0. The closed curves y, and y, areobtained by taking sections transverse to the tube of flow flines. (b) Dissection of the tube shown in (a) used to obtain (1.9.35).
Figure 1.37 If y, is given by r = r(u), then r(u) = (q,(u), 0, p,(u), p,(u), t(u)), where p2(u) is determined by H(q, p) = h,. The curve, y,, obtained by projecting y, onto t = 0 (see (a)), is the image of y, under the Poincark map P (see (b)).
1.9 Hamiltonian flows and Poincarh. maps 5 1
the tube as shown in Figure 1.36(b), we observe that
curlvdS=0= /y, ~ . d r  / ~ ~ v  d r , (1.9.35)
where dr = (dq, dp, dt). Thus
and it follows that p dq  H dt is invariant under the flow. With the aid of differential two forms (see Arnold, 1968, pp. 2346), we can
obtain Stokes' Theorem in five dimensions and derive the corresponding result for 2F systems; namely
p1 dql + pd dq2  H dt = PI dql + p2 dq2  H dt, (1.9.37)
Figure 1.38 Some typical orbits ofthe HCnon map (1.9.40) for cos a = 0.8. Two fixed points can be seen: one elliptic (see $6.5) and one saddlelike. What appear to be closed curves are each the orbit of a single point, i.e. the orbit is confined to what is topologically an invariant circle. For small numbers of iterations of (1.9.40) individual points of these orbits can be distinguished moving around the origin (d. Exercise 1.9.9). As the number of iterations increases, the plotted points merge into what looks like a closed curve. Individual orbit points are more apparent in the vicinity of the saddle point. (After Hknon, 1969.)
52 I Diffeomorphisms and Jows 1.9 Hamiltonian jlows and Poincare maps 53
where y , and y, are closed curves bounding a tube of the flow in the fivedimensional extended phase space with coordinates (q,, q,, p,, p,, t). Now let y, consist entirely of points such that H = h,, q, = t = 0. Suppose we follow the lines of the flow (p: until we return to q2 =O. Although H remains at h, and q2 returns to zero, the
Figure 1.39 A selection of plotted orbits of (1 9.40) for cos a = 0.4. Orbits of points near to the saddle point become highly irregular. Successive iterates still move around the fixed point at (0.0) but they are no longer confined to a closed curve. Instead they appear to spread over a twodimensional region in an erratic manner. Eventually, they are pulled away along the unstable manifold of the saddle and, left to themselves, will cause an overtlow error in the computer doing the plotting. On the other hand, orbits of points near the origin still appear to be confined to invariant circles. Between these extremes, a new feature called an island choin can be seen. The 'islands' themselves are formed around the points ofan elliptic periodic orbit, here of period six. The 'straits' between successive islands contain a hyperbolic periodic orbit also of period six. The orbits of points near to the elliptic periodic points move from island to island, returning to an invariant circle surrounding the initial elliptic point at every sixth iteration. Some information to help the reader to observe island chains is given in Exercise 1.9.9. (After Htnon, 1969.)
time required to reach q, = 0 will, in general, be different for different points of y,. Thus, in extended phase space, points of the image, y,, of y, do not all have the same t coordinate. Let us put y, and y, defined in this way into (1.9.37). Since H and q, are constant on both curves, we have
for i = l , 2 , so that (1.9.37) becomes
J PI dql = I PI dq1 = PI 4 1 , (1.9.39) 71 72
where f , is the projection of y, onto t =O. Now 7, is the image of y, under the
Figure 1.40 Analogous plots to those shown in Figure 1.39 but with cos a = 0.24. Observe that a fivefold island chain is the dominant feature here. In fact, (see 56.5) island chains of all periods occur but only a few are easily visible. The orbits looking like separatrices of the hyperbolic periodic points are deceptive (see Figure 1.41). (After Htnon, 1969.)
54 I Difleomorphisms and flows
Poincart map P (see Figure 1.37). Hence P is an areapreserving map on the section H(q, p) = h, and q, = 0 in the phase space of the system.
Numerical experiment has shown that Poincart maps constructed in the manner described above exhibit complicated dynamics (Htnon, 1983, pp. 8495; Lichten berg & Lieberman, 1982). This complexity is a feature of areapreserving maps of the plane and it is typified by the quadratic mapping of Htnon: namely
x,, , = X , cos m  y, sin a + x: sin m,
y,+,=x,sina+y,cosax:wsm,
where a is a real parameter and t E Z (see Htnon, 1969). This map is not constructed
Figure 1.41 Two orbits of (1.9.40) for ws a = 0.22. The first is the orbit of a point near an island centre giving invariant circles around the five elliptic periodic points. In the present context, it serves only to indicate the position of the islands. The remaining points are all generated by iterating a single initial point. Once again, the iterates spread out, in a stochastic manner, over a twodimensional region in the neighbourhood ofwhat appeared to beseparatrices in Figure 1.40. (After HCnon, 1969.)
1.9 Hamiltonian flows and Poincare maps 55
as the Poincart map of a Hamiltonian system. Instead it represents the most general quadratic planar map that is areapreserving and has a pure rotation for its linear part. Some striking features of the dynamics of (1.9.40) are illustrated in Figures 1.381.42 but the reader cannot do better than to consult Htnon's excellent review (1983) for more details. Figures 1.381.42 show invariant circles, islands chains, chaotic orbits and their repetition on all scales. All this leads to a picture of immense complexity that is by no means fully understood. We will return to such matters in Chapters 3 and 6.
Figure 1.42 The result of magnifying a detail of Figure 1.40 containing one of the hyperbolic periodic points. A twodimensional orbit like that shown in Figure 1.41 is apparent. However, not only are more island chains visible around the fixed point (0, O), but also analogous islands can be seen around the adjacent elliptic periodic points. As we shall see, if these islands were again magnified, then we should find more twodimensional orbits and more island chains and so on. Thus the complexity of the map is repeated on all scales. (After HCnon, 1969.)
58 1 DiJfomorphisms and flows
1.4.6 Use the PoincarkBendixson Theorem to show that the Van der Pol oscillator
has at least one stable limit cycle for suficiently small values of E.
15 Conjugacy
1.5.1 Showthat: (i) j(x) = 2xand g(x) = 8xare topologically conjugate on W but not differentiably
conjugate; (ii) f(x)=2x and g(x)= 2x are not topologically conjugate by showing
conjugacy preserves orientation of a map; (iii) j(x) = 2x and g(x) = $x are not topologically conjugate by investigating the
nature of the fixed point at the origin in the two cases.
1 .5.2 Prove that cp: XHX'"+~ , neN, is a topological conjugacy of the diffeomorphisms j(x) = 2x and g(y) = 22"+1y on R. Why is there no differentiable conjugacy when 11 > 0.
1.5.3 Let f, g: Ra t Re be Ckconjugate (k 2 1) difleomorphisms by h: R" t IR" and suppose f(0) = 0. Prove that the Jacobian matrices of f at 0 and g at h(0) are similar. What does this imply about the eigenvalues of Df(0) and Dg(h(O))?
Show that j(x) =ax and g(x) = px are not Ckconjugate for a # 1. When are j and g not COconjugate?
1.5.4 Let j, g be diffeomorphisms on R given by
f(x) = x + sin x (E1.10)
and
where
k€Z+. Prove that j and g are not topologically conjugate for any k.
1.5.5 Find the number of period2 points of the diNeomorphisms f and g of the circle S1, where the lifts are:
f(x) = x + 0.5 + 0.1 sin 2nx; (E1.13)
y(x) = x + 0.3 $0.1 sin 2nx. (E1.14)
Show that j and g are not topologically conjugate.
1.5.6 Prove that the two linear systems x =Ax, y = By, r , y ER", have flows which are linearly conjugate if and only if the matrices A and B are similar.
1.5.7 Let Q, $ be flows on Wn and suppose that Q has a Sxed point at the origin. If Q
and (G are CLconjugate, k > 0, by h: W" r R", show that the vector fields X and Y of Q and $ are such that the matrices DX(0) and DY(h(0)) are similar. Compare your answer with that of Exercise 1.5.6 and comment on why the converse of the above is not true in the nonlinear case.
1.5.8 Find the rotation number of the circle homeomorphism f with lift f R t R given
Exercises
by: (i) fix) = x + $;
(ii) f ( x ) = x 3 + ~ , 0 ~ x < 1 , ~ ( x + + ) = ~ ( x ) + 1; 1 . (iii) T(x) = x + t +  sin 2nx;
271 by locating fixed or periodic points off.
6 1.6 Eguivalence of flows 1.6.1 Show that flows
0 6 0  1 ex~(Jt), J = ( p ,.) and exp(Jor), Jo =
0)
are topologically equivalent. Hence prove that all linear flows, exp(At), for which A has pure imaginary eigenvalues are topologically equivalent to exp(Jot). Why are these flows not all topologically conjugate?
1.6.2 It is often easy to recognise why two flows are not topologically equivalent by noting key features in their phase portraits. Describe, for each diagram in Figure E1.1, a distinguishing feature preserved by topological equivalence which is not shared by the others. Explain your answers. Why is the invariant circle in (b) not a limit cycle?
1.6.3 Show that the topological types of a saddle and node are different by considering the separatrices of the saddle.
1.6.4 Consider the system x = X,(x), a€ R,
i = 1 , j = a (E1.16)
on R2 and the flow on T 2 induced by the map n: R2 + T' where n(x, y)= (x mod 1, y mod 1). Show that the phase portraits of the system (E1.16) for a (a) rational and (b) irrational are not topologically equivalent.
1.6.5 Let Q and Q' be topologically conjugate flows on the manifold M and $ and $' be topologically conjugate flows on the manifold N. Prove that the product flow
1 Diffeomorphisms and flows
Q x $is topologically conjugate to Q' x f on M x N. Show that this result does not extend to topological equivalence of flows by considering flows on M = N = S1.
1.6.6 The topological types of linear flow, exp(At), given in Figure 1.26 can be subdivided into algebraic types. Sketch phase portraits for exp(At) when: (a) [Tr A]' > 4 Det A (nodes); (b) [Tr A]' = 4 Det A (improper nodes); and (c) [Tr A]' < 4 Det A, Tr A # 0 (foci). Modily Figure 1.26 to show these algebraic types on the (Tr A, Det A)plane.
1.7 Poincar6 maps a d suspensions
1.7.1 Show that a map f: W + R of the form f(x)= ax, aER, has a fixed point at x = O and that it is stable or unstable according as (a( < 1 or la1 > l respectively.
Consider the timeone map rpl of the flow of i = x x2. Deduce that cp, has fixed points at x = 0 and x = 1. Obtain the linear approximations to q l at these points and deduce their stability. Sketch the behaviour of the map cp,. Compare your results with the phase portrait of the flow.
1.7.2 Show that flows with fixed points do not satisfy the requirements for.the existence of a global cross section. In what sense can the system i = r  r', 0 = 1, be said to have a global section, S = {(r, 0)lr 2 0, 0 = 0), on which the Poincark map determines the dynamics?
1.7.3 Let y be a periodic orbit of a flow Q. Suppose S,,S, are distinct local sections for y (see Definition 1.5.2) at x,, x , ~ y such that S, = cp,,(S,). Let P,, P, be the corresponding Poincark maps on S, and S,. Prove that for suitable neighbourhoods of x1 and x,, there is a C1conjugacy between maps P, and P,.
1.7.4 Suspend the diffeomorphism f: [O,1] + [0, I] described by f(x) = i(x + x2) to obtain a flow on a surface. What is the surface? Describe the behaviour of the flow.
1.7.5 Draw diagrams to illustrate the suspended flows of the diffeomorphisms (a) f : I + I , I = [  l , l ] , x w  j x + + x 3 , (b) j: S1 + S1, exp(2nix)exp(2nix).
(E1.17)
1.7.6 Show that the flow Q,(x, y) = ((x + t)mod 1 , y t at) on the cylinder (x mod 1, y)l(x, y ) ~ R2} is the suspension of the diffeomorphism f: W + R given by .f(y) = Y + a.
1.8 Periodic nonautonomous systems
1.8.1 Let x = X(x, t), (x, t ) ~ W" x R, be a periodic differential equation with X(x, t) = X(x, t + T) for some T > 0. Show that the transformations t' = (2nlT)t and X1(x, t') = (T/2n)X(x, t) give a 2nperiodic system dxldt' = X 1 ( x , t').
1.8.2 Prove that if x = C(t) is a solution of x = X(x, I), with X(x, t + T) = X(x, t) and (x, t )sRn x R, then x = <(t + T) is also a solution. Show that such systems can have nonperiodic solutions by considering 2 = (1 +sin t)x with (x, ~ ) E R x R.
1.8.3 (a) Show that the set ofsolutions ofx = A(t)x, (x, t ) ~ W x R, form a vector space. (b) Let {,(t), i = I , . . ., n + l be a set of solutions. Show that there exist
Exercises
n t l
a,, . . ., a,, ,, not all zero, such that aiCi(0) = 0. Use uniqueness of solution i = 1
n t l
to show that aiCi(t) E 0. i = 1
(c) Show that the vector space of solutions is ndimensional.
1.8.4 Define the state transition matrix ~ ( t , to) of the system x = A(t)x, (x, t ) ~ Wn x R, A(t + T) = A(t). Prove that (i) cp(t + T, to + T) = ~ ( t , to),
(ii) ~ ( t , 0)I = ~ ( 0 , t). (E1.18)
Use these results to show that
~ ( t + T, t) = ~ ( 0 , t )  ' dT , O)Q(O, t). (E1.19)
What does this imply about the family of Poincare maps P,: C, + C,, 0, E [O, 2x1 of the system
x=A(O)x, 9 = 1,
where A(O + 2n) = A(@, and C,o = {(x, 0,)lxaR2)?
1.8.5 The state transition matrix is useful where A is independent of t or aperiodic.
(a) Verify that t,(t) = ("P;lt3 and t2(t) = (exoyn,t)) are linearly independent . .
solutions of x = Ax, A = (: :). Hence find ~ ( t , to) for this system.
(b) Verify that {,(t) = (ex:r)) and t2(t) = (n$)) are a basis for the solution
space of x = A(t)x with A(t) = . Hence construct ~ ( t , 0)
and find x(t) given that x(0) = . (3 1.8.6 Let ~ ( t , to) be the state transition matrix of the system x = A(t)x and define W(t) =
Det(cp(t, 0)). Prove (i) ~ ( t + h, 0) = ~ ( t , 0) t hA(t)~(t, 0) + 0(h2); (E1.21)
(ii) W(t + h) = Det(1 t hA(t) + 0(h2)) W(t)
where li, i = I , . . ., n are the eigenvalues of A; (iii) f i ( t ) = Tr(A(t))W(t).
1.8.7 Find solutions for the system
(El .22)
and hence, or otherwise, obtain the period advance map p(2n, 0) = P. Calculate the eigenvalues of P and determine the stability of the null solution of (E1.24).
1.8.8 If ~ ( t , to) is the state transition matrix of the system
1 Difl'eomorphisms and flows
show that
x = A(t)x + B(t) has solution
whenx=x,at t=t,. Find the solution of
when x(0) = x,.
1.9 Hamiltonian flows and Poinear6 map
1.9.1 Find the phase portraits of the flows in R2 with Hamiltonian: (a) H(x,, x,) = x: + x:; (b) H(x,, xz) = x: + x:; (E1.29) (c) H(x,,x~)=x:+x':x:.
1.9.2 Sketch the phase portraits for Hamilton's equations when H(x,,x,)=x: + x: + px2 and
(i) p < 0; (ii) p = 0; (iii) p > 0.
1.9.3 Prove that the fixed point of a planar flow with a quadratic Hamiltonian is generically either a centre or a hyperbolic saddle.
1.9.4 Show that, for p >0, the tIamiltonian vector field given by H = pr2 + r4 + rs cos 50, where (r, 0) are plane polar coordinates, has 11 fixed points consisting of 6 centres and 5 saddles. Show that the separatrices of the saddles form achain of 5 islands around the origin such that each island containsacentre.
1.9.5 Prove that a change of variables on R2 is symplectic if and only if it is orientation and areapreserving. Show that: (i) the change from Cartesian to plane polar coordinates, (x, y ) ~ (r, B), is not
symplectic; (ii) the transformation (x, y ) ~ (7, B), where r = r2/2, is symplectic. Illustrate the fact that symplectic transformations preserve the form of Hamilton's equations by applying the transformations in (i) and (ii) to the system.
1.9.6 Let he L(R4) be given by h(x) = Px with
where A, B, C, D are 2 x 2 matrices. Show that h is symplectic if and only if
DTA  BTC = I, DTB = BTD and CTA = A ~ C , (E1.32)
Exercises
1.9.7 Show that the VolterraLotka equations
.i = (u  by)^, jr =  (c  jx)y (E1.33)
do not form a Hamiltonian system. Show that the change of variable x = exp(q), y = exp(p) allows (E1.33) to be written as a Hamiltonian system with Hamiltonian
Why can it beconcluded (without calculation) that the transformation (q, p ) ~ (x, y) is not symplectic?
1.9.8 (a) Find actionangle variables for the Hamiltonian system
by using I = xZ + y2. (b) Consider the pendulum equations
i = y , y=sinx (E1.36)
and show that the transformation Y : (x, y ) w ( [ ,a), where I = (y2/2)  cos x and 0 is the polar angle, has Jacobian 1 + O(1). Show that the transformed vector field is
i = o , e= t+o( l r l ) . (~1.37)
In view of Exercise 1.9.5 what do these results imply about Y?
1.9.9 Write a computer program to plot orbits of the HCnon areapreserving map (1.9.40). (a) For cos 1=0.8, plot several orbits with initial points (x, 0) for xe(0, f].
Compare your results with Figure 1.38. (b) For cos a = 0.4, plot orbits with initial points (x, 0) for x 6 (!,I]. Observe a
sixfold island chain and chaotic orbits.
where I is the 2 x 2 unit matrix. Hence construct a counterexample to show that Det(Dh(x)) = 1 does not imply h is symplectic.
118 2 Local properties of Jows and [email protected]
2.8.6 Use blowingup techniques to determine the topological types of the singularities at the origin in the following systems:
(a) i = y + x 3 , y = x 3 ; (b) I = x2  y2, y = 2xy.
Explain why the technique fails to determine the topological type of the singularity at (x , y) = 0 in the system .i = + x 3 , j =  x 3 .
Structural stability, hyperbolicity, and homoclinic points
In applications we require our mathematical models to be robust. By this we mean that their qualitative properties should not change significantly when the model is subjected to small, allowable perturbations. If we wish to make these ideas more precise, then we must have some class of perturbations in mind and some way of deciding when they are small. From a theoretical stand point, this means that we regard our model as a member of some chosen space, Y, of dynamical systems to which we attach an appropriate metric. It is then possible to give meaning to the idea that a perturbation is 'close' to the original model. A dynamical system whose topological properties are shared (in a sense that must be properly defined) by all sufliciently close neighbouring systems is said to be structurally stable.
Structural stability, like hyperbolicity (see s 2 . 1 and 2.2), is a property of individual dynamical systems and we can ask if this property is, in some sense, typical of the elements of the space 9 The subset of all structurally stable systems is open. This follows directly from the definition of structural stability itself. Clearly, every structurally stable system lies in an open set, each element of which is also structurally stable. Thus the subset of structurally stable systems is a union of open sets and is therefore open itself. It follows that a structurally stable system cannot be approximated arbitrarily closely by structurally unstable systems. However, in some cases the subset of structurally stable systems can be shown to be dense in the space 9 This means that every structurally unstable system can be approximated arbitrarily closely by structurally stable systems. In such cases, we say that structural stability is a generic property of 9 In general, a property is said to be generic if it is shared by a residual subset of the space of systems involved. Such a subset is a countable intersection of open dense sets. We will, however, encounter only open dense sets in this chapter.
In 03.1, we show that a linear flow is structurally stable if and only if it is hyperbolic and that hyperbolicity of flow is a generic property of linear trans formations on Wn. Thus the structurally stable linear flows are characterised by a single, hyperbolic fixed point at the origin and structural stability of flow is a generic property of linear transformations.
120 3 Structural stability, hyperbolicity and homoclinic points
Studies of flows on twodimensional compact manifolds (see 83.3) show that the structurally stable systems have nonwandering sets which are characterised by hyperbolic fixed points and closed orbits, together with the global requirement that no saddle connections occur. Structural stability is, once again, a generic property of these flows.
The above findings led to two conjectures about flows on manifolds of dimension n 2 2 :
(i) structural stability is a generic property of such flows;
(ii) the structurally stable flows are characterised in the same way as flows on twomanifolds.
Neither of these conjectures is correct. A counterexample to the first was given by Smale (1966). We do not discuss this example here; the interested reader can consult Arnold & Avez, 1968, pp. 196200. In g 3 . 4 and 3.5 we describe two types of diffeomorphism on twomanifolds that are structurally stable but cannot be characterised in the manner described in (ii). Remember (see $ 1.7) these diffeo morphisms correspond to flows in three dimensions by suspension. The Anosov automorphisms of the twotorus (see 4 3.4) and the horseshoe direomorphism of the sphere (see $3.5) have complicated nonwandering sets and to characterise them we must extend our notion of hyperbolic sets beyond fixed points and closed orbits (see 43.6). The systems referred to in (ii) are now known as MorseSmale systems (see Nitecki, 1971).
The horseshoe diffeomorphism also plays a central role in our understanding of the complex dynamics described in 41.9. In particular, it can be shown that it has orbits that behave in a random or chaotic way (see g3.5.2 and 3). In # 3.6 and 3.7 we explain how this behaviour is related to the 'twodimensional, chaotic orbits' associated with hyperbolic fixed or periodic points of some planar maps (see Figures 1.3942). The occurrence of homoclinic (or heteroclinic) points is the central feature of this discussion. They arise when the stable and unstable manifolds of a hyperbolic point (or points) intersect transversely and it can be shown that the map must then contain embedded horseshoes.
3.1 Structural stability of linear systems
Let L(R") be the set of real linear transformations of R" to itself. Define the norm n
of an n x n matrix A =.[a,,] to be IlAll = An Eneighbourhood of A is given i . j= 1
by N,(A) = {BE L(Wn)J IIB  All < E). Each BE N,(A) is said to be &close to A. We are now able to give a formal definition of structural stability for linear flows and diffeomorphisms on R".
3.1 Structural stability of linear systems 121
Definition 3.1.1 A linear Jlow, exp(At): Rn r Rn, (or difleomorphism, A) is said to be structurally stable in L(Rn) ) there is an Eneighbourhood of A, N,(A) G L(Rn), such that, for every B E N,(A), exp(Bt) (or B) is topologically equivalent (conjugate) to exp(At) (or A).
The following result shows that, in this linear case, the structurally stable systems can be completely characterised.
Proposition 3.1.1 A linear Jlow or diJeomorphism on Rn is str~rcturally stable in L(Rn) if and only if it is hyperbolic.
Proof. A linear flow exp(At) is hyperbolic if all the eigenvalues of the matrix A have nonzero real parts (see Definition 2.1.2). The eigenvalues of any &close matrix B differ from those of A by terms O(E) (see Exercise 3.1.1). Thus, by making E sufficiently small we can ensure that the eigenvalues of B are near enough to those of A for their real parts to be nonzero. Moreover, A and B will then have the same number, n,(n,), of eigenvalues with negative (positive) real parts. Theorem 2.1.2 then implies that exp(At) and exp(Bt) are both equivalent to the flow of x = x, y = y, where X E Rns and y E Wnu. Thus A is structurally stable.
Conversely, suppose the flow exp(At) is not hyperbolic. Then A has at least one eigenvalue with zero real part. However, B = A + EI is hyperbolic for almost all E # 0 and can be made arbitrarily close to A by taking E sufiiciently small. Thus, the nonhyperbolic flow exp(At) is not structurally stable. Hence, if a linear flow is structurally stable it must be hyperbolic (see Figure 3.1).
The proof of Proposition 3.1.1 for diffeomorphisms follows similar lines; it is considered in Exercise 3.1.2.
It is important to note that Definition 3.1.1 specifies a space of systems (L(Wn)) to which the perturbations must belong. Whether or not a given system is structurally stable depends on the choice of this space. For example, let CL(R2) E L(R2) be the subspace of linear transformations with pure imaginary, nonzero eigenvalues. If A E CL(W2), it is structurally stable in CL(W2) but structurally unstable in L(W2). Clearly, if BE CL(R2) and is Eclose to A, then B has pure imaginary eigenvalues that are numerically close to those of A. Thus, the flows exp(At) and exp(Bt) are both of centre type and therefore topologically equivalent (see Exercise 1.6.1). Hence A is structurally stable in CL(W2). Of course, A€CL(R2) is not structurally stable in L(R2) (see Figure 3.2). This follows from Proposition 3.1.1 because A is not hyperbolic.
The relationship between hyperbolic and structurally stable flows in L(Rn) allows us to show that 'structural stability of flow' is a generic property of linear transformations. Let SF(Rn) c L(Rn) denote the set of linear transformations which give rise to structurally stable flows on Rn.
124 3 Structural stability, hyperbolicity and homoclinic points
Proposition 3.2.1 Let X E Vecl(U) have a hyperbolic sitzgularity at x = x*. Then there exists a neighbourhood V of x* in U and a neighbourhood N of X in Vecl(U) such that each Y E N has a unique hyperbolic singular point y * ~ V. Moreover, the linearised flow, exp(DY(y*)t) has stable and unstable eigenspaces of the same dimension as exp(DX(x*)t).
Notice the perturbed fixed point y* does not, in general, coincide with x*. However, given any 6 > 0, N can be chosen such that (y*  x*( < 6 for all YEN. Furthermore, by using Proposition 3.2.1 in conjunction with Hartman's Theorem (Theorem 2.2.3) we can deduce that there are neighbourhoods of x* and y* on which X and Y are topologically equivalent. Proposition 3.2.1 gives equality of dimension of the stable eigenspaces of the linearised flows exp(DY(y*)t) and exp(DX(xS)t). It follows that these linear flows are topologically equivalent (see Theorem 2.1.2). The Hartman Theorem states that there is a neighbourhood of x* (y*) on which the flow of x=X(x) (y =Y(y)) is topologically conjugate to exp(DX(x*)t) (exp(DY(y*)t)). Thus the local, Coequivalence of the flows of X and Y follows from
where Q, and #t are the flows of X and Y respectively. U:. and U!* G U denote the neighbourhoods on which Hartman's Theorem is valid. There is, therefore, a sense in which Q,: U + Wn is structurally stable in a neighbourhood of x*. Namely .that for every pair (Y, y*) with Y E N there is a neighbourhood Ll;. G U of y* such that +,(U,H. is Coequivalent to q,lU$. Thus, one might say that p, is locally structurally stable at x*. Alternatively we can observe that (3.2.2) means that the topological type of the fixed point (see opening paragraph of Chapter 2) is preserved under all sufficiently small C1perturbations and say that the type of fixed point is structurally stable.
Hyperbolic fixed points of diffeomorphisms also persist under sufficiently small C1perturbations. Let Diffl(U) be the set of C1diffeomorphisms f: U(sIWn) + R" with the C1norm. Then the following result parallels Proposition 3.2.1 for flows.
Proposition 3.2.2 Let x* be a hyperbolic fixed point of the diffeomorphism f: U + R". Then there is a neighbourhood V of x* in U and a neighbourhood N off in DiB1(U) such that every g~ N has a unique hj~perbolic fixed point y * ~ V of the same topological type as x*.
Proposition 3.2.2 allows us to show that hyperbolic closed orbits of flows on R" are structurally stable.Let the flow 9,: U + R", with vector field XEVecl(U), have a hyperbolic closed orbit y. Then, its Poincark map P defined on a local section S a t X*EY belongs to DiB1(S). Moreover, x* is a hyperbolic fixed point of P. For sufficiently small E, every EC1perturbation of P has a hyperbolic fixed point of the same topological type as x*, by Proposition 3.2.2. Hence, every sufficiently
3.3 Flows on twodimensional manifolds 125
small C1perturbation of X gives rise to a flow with a hyperbolic closed orbit of the same topological type as y.
3.3 Flows on twodimensional manifolds
The characterisation of structurally stable flows on twomanifolds provides a good illustration of the technical complications that can arise when we try to extend the local discussion of $3.2 to include global phenomena. Recall that the results obtained in $3.2 depended on the existence of sufliciently small neighbourhoods of the unperturbed and perturbed fixed points on which topological equivalence could be established. These neighbourhoods were sufficiently small subsets of the open set U on which X was defined and therefore we were not involved with the behaviour of X on the boundary, aU, of U. Indeed, as far as the local discussion was concerned we had not considered whether or not X had a natural extension to aU. Moreover, since we were involved with perturbations differing from X only on sufficiently small neighbourhoods of the singular point x*, the question of whether JIX  YJ1, was finite for all X, Y in Vecl(U) was not relevant. Complications of this kind cannot be ignored if precise statements are to be made about global structural stability and they give rise to a number of technical conditions in the resulting theorems.
Let us begin by considering vector fields on R2. We can ensure that (IX  Y(I, is defined for all X, Y by restricting the discussion to compact subsets of R2. Therefore, let D2 = {xER~IIxI < 1) and let aD2 denote its boundary. To ensure that the vector fields involved are well defined on aD2, we will assume they are defined on an open set U containing DZ and then take their restriction to the unit disc. Let Vec'(DZ) be the set of all C1vector fields, X, defined in this way equipped with the C1norm
Definition 3.3.1 A vector Jield X in Vec'(D2) is said to be structurally stable if there exists a neighbourhood, N, of X in Vec'(DZ) such that the flow of every Y in N is topologically equivalent to that of X on D2.
The above precautions are not enough to focus attention on the structural instabilities occurring in the interior of D2. Unfortunately, instabilities associated with the behaviour of the vector fields at the boundary of the disc can still occur. In the absence of further constraints, vector fields that are tangent to aDZ are still present in Vec'(D2). Topologically distinct C1perturbations of such vector fields are illustrated in Figure 3.3. These unwanted structural instabilities can be excluded by confining the discussion to those vector fields that are transverse to aD2. Clearly, a vector field that satisfies this requirement is a member of one of two disjoint
126 3 Structural stability, hyperbolicity and homoclinic points
subsets of Vecl(DZ): it either points into or out of D2 at every point of aD2. Let Vec,',(D2) be the set of vector fields defi ned on the disc D2 in the manner described above and such that X(x) points into D2 for every x in aD2. Then the following theorems generalise Propositions 3.1.1 and 3.1.2 for linear vector fields.
Theorem 33.1 (Peixoto) Let X belong to Vec;,,(D2). Then X is structurally stable i f and only i f its jlow satisfies:
(i) all fixed points are hyperbolic; (ii) all closed orbits are hyperbolic; (iii) there are no orbits connecting saddle points.
Notice that items (i)and (ii)in Theorem 3.3.1 simply ensure local structural stability of the fixed points and closed orbits in the flow of X. It is really only item (iii) that involves a global property of the flow.
Theorem 33.2 The subset of vector fields in Vec;,,(D2) that are structurally stable is open and dense in Vec;(D2).
Obviously, parallel results could be stated for those vector fields that point out of D2 at every point of 8 ~ ~ . Indeed, the flows of vector fields in V e c i ' , ( ~ ~ ) and Vec:,,(D2) are in 1:l correspondence by time reversal.
By stereographic projection (see Figure 3.4(a)), the unit disc D2 is homeomorphic to a closed cap, C2, based on the south pole of the sphere S2. The boundary condition on aDz can then be replaced by considering flows on S2 with a single repelling fixed point in S2\C2. This is conveniently placed at the north pole (see Figure 3.4(6)). Provided that this additional fixed point is hyperbolic, a structurally stable vector field on D2 is also structurally stable on S2. This follows from the work of Peixoto (1962) who extended Theorems 3.3.1 and 3.3.2 to vector fields on twodimensional, compact manifolds. Let M be a twodimensional, compact manifold without boundary and let Vecl(M) be the set of C1vector fields on M
Figure 3.3 The vector field in Vec1(D2) with a tangency at x o ~ a D 2 in (a) is not structurally stable as the perturbation (b) shows. In (b), an orbit leaves D2 with increasing time.
3.3 Flows on twodimensional manifolds 127
with the C1norm. This norm is defined by imposing the C1norm on each of the charts of a finite atlas for M. Then, Peixoto's result may be stated as follows.
Theorem 3.33 (Peixoto) A vector field in Vecl(M) is structurally stable i f and only i j its flow satisfies:
(i) all fixed points are hyperbolic; (ii) all closed orbits are hyperbolic; (iii) there are no orbits connecting saddle points; (iv) the nonwandering set consists only of jixed points and periodic orbits.
Moreover, i f M is orientable the set of structurally stable C1vector fields forms an open dense subset of Vecl(M).
Here orientable simply means that two distinct sides of M can be recognised. The sphere, torus, pretzel, etc. are all examples of orientable manifolds.
The statement of Theorem 3.3.3 contains an additional condition (iv) that does not appear in Theorem 3.3.1. Its role can be illustrated as follows. Consider the irrational flow on the toms T2. This flow satisfies items (i)(iii) vacuously. There are no fixed points, closed orbits or saddle connections. However, it fails to satisfy item (iv) because the nonwandering set is the whole of T2. Therefore, the irrational flow is not structurally stable on T2. Clearly, there exist EC1close rational flows for which every orbit is closed. Some examples of structurally stable flows on S2 and T2 are shown in Figure 3.5.
It should perhaps be noted that, since M is compact, flows on it can only have finitely many fixed and periodic points if they are all hyperbolic. By the Hartman
Figure 3.4 (a) The stereographic projection from the north pole N of the sphere. The circle Y projects onto the boundary of D'. (b) A vector field in a neighbourhood of N which cuts Y transversely.
N
128 3 Structural stability, hyperbolicity and homoclinic points
Theorem, the flow in the vicinity of a hyperbolic fixed point is topologically conjugate to that of its linearisation. The latter has an isolated fixed point at the origin, hence hyperbolic points must be isolated. This means that fixed points cannot accumulate at a hyperbolic fixed point. Consequently, only a finite number of fixed points can occur on a compact manifold if they are all hyperbolic.
We might try to extend the discussion of structural stability to noncompact sets such as the whole plane. Of course, a finite C1norm can no longer be guaranteed but an obvious approach is to consider restrictions to compact subsets. Clearly, if a vector field is structurally unstable on any compact subset of the plane then it can be deemed structurally unstable on the whole plane. This approach does have practical merit since in applications we are rarely involved with variables passing to infinity, rather they become very large but finite. The following example illustrates this idea.
Example 3.3.1 Show that the vector field, X, of the diflerential equation
is not structurally stable on any compact subset of the plane with the line segment joining the singular points of X in its interior.
Soltrtion. The system (3.3.2) has saddle points at x* = (0,O) and y* = (2,O). On the xaxis, y = 0 and so there is an orbit connecting these hyperbolic fixed points. The phase portrait is shown in Figure 3.6(a). Let D be any compact set containing the common separatrix of x* and y*. Notice that the stable manifold of x* is the line x = 0 while the line x = 2 is the unstable manifold for y*. This means that
Figure 3.5 Exanlples of structurall~ stable phase portrai!~ on the sphere and !orus. ( a ) 0 = sin 0 ,@ = 0; (6 ) 0 = sin 28,@ = 1 ; (c) 0 = 1,4 = sin cp; (d) O=sin 28, 4 = 1.
3.3 Flows on twodimensional manijdds 1 29
there are points x in the boundary of D for which X(x) points both into and out of D. Therefore, we are (technically) not able to apply Theorem 3.3.1. However, consider the oneparameter family of systems
The vector field of (3.3.3) can be made EC1close to that of (3.3.2) on any compact subset D of the type described above by taking p sufficiently small. The phase portrait for (3.3.3) with p > 0 is shown in Figure 3.6(b). The points (0,O) and (2,O) are saddle points for all real p. However, for nonzero p, Q $ 0 on the xaxis between these points. Moreover, the stable separatrix at (2,O) is tangent to y =+fix. Therefore, for p > 0 there is no saddle connection between the fixed points. Hence, the flows for p = 0 and p > 0 are topologically distinct. The vector field in (3.3.2) is consequently structurally unstable on every compact subset of the plane containing the saddle connection.
It is perhaps worth noting that, in view of the role played by the boundary condition on aDZ in Theorem 3.3.1, we can say that a vector field that fails to satisfy the conditions (i)(iii) will certainly be structurally unstable independently of the boundary condition on aD2. Of course, the converse is not true unless the vector field is transverse to the boundary.
Example 3.3.1 illustrates a useful way of giving meaning to structural instability on the plane. Some other examples are shown in Figure 3.7. However, we must not be misled into believing that if a vector field is structurally stable on arbitrarily large compact subsets of the plane then it is structurally stable on the whole plane. The following is a counterexample to this erroneous conjecture.
Example 33.2 Show that there are arbitrarily large compact subsets of the plane on which the system
Figure 3.6 Phase portraits for ( a ) (3.3.2), (b ) (3.3.3) with y >0.
I
130 3 Structural stability, hyperbolicity and homoclinic points
is structurally stable. Verify that the topological type of (3.3.4) on the whole plane is changed by the addition of the perturbation (0, p ) to (2, j ) , however small the value of y # 0.
Solution. Figure 3.8(a) shows j as a function of y. It follows that (3.3.4) has fixed points at (j~, y ) = (0 , p), p e Z . These fixed points are alternately stable nodes and saddle points as shown in Figure 3.8(b). Suppose D is a compact subset of R2 whose boundary intersects the yaxis at (0, y,), (0, y,,) with y, < y, and y,, y, # p for any integer p. If D is such that j (y , ) > 0 and j (y , ) < 0 then (3.3.4) is structurally stable on D by Theorem 3.3.1. Clearly, arbitrarily large D can be constructed in this way.
Consider the family of vector fields defined by x = X,(x), where
Figure 3.7 Some examples of structurally unstable phase portraits on R2. In each case the structural instability is apparent in the restriction of the flow to a compact subset of the plane.
3.3 Flows on twodimensional mangolds 131
The vector field of (3.3.4) is X, and
2 ax; ax;  C  y . (3.3.6) i.j=l ax, axj I I)
Thus X, can be made eC1close to X, on the whole plane. However, X,, p Z 0, has only finitely many fixed points (see the broken curve on Figure 3.8(a)). Hence, the flow of (3.3.5) is topologically distinct from that of (3.3.4) for any nonzero value of p.
Notice that the topological type of the flow of X, in Example 3.3.2 changes when p departs from zero, even though it has only hyperbolic fixed points and no saddle connections. Since the plane is not compact, infinitely many hyperbolic fixed points can occur. If we wish to maintain contact with Theorem 3.3.1, we must specify a boundary condition at infinity. However, if we require that all vector fields point inward on the boundaries of all sutliciently large discs, we are led back to Theorem 3.3.3 via stereographic projection. In other words, we can obtain structurally stable flows on R2 from structurally stable flows on SZ, however, they will have only finitely many hyperbolic fixed points and their behaviour at infinity will correspond to having a hyperbolic fixed point at the north pole of the sphere (see Figure 3.9).
Figure 3.8 Phase portrait of (3.3.4) in relation to plot of j, as a function of y. All zeroes of y are simple. The lixed point (x*, y*) is a saddle if dj/dyl,. > 0 and a node if dj/dy(,. <0. The broken curve in (a) is a plot of j versus y for (3.3.5). The decay of the amplitude of the oscillations in jr means that there are no fixed points for exp(y2) < lyl, i r . for y2 > In(l~l).
132 3 Structural stability, hyperbolicity and homoclinic points
3.4 Anosov diffeomorphisrns
The relationship between flows on nmanifolds with global sections and diffeo morphisms on manifolds of dimension n  1 (via Poincark maps) was discussed in 1.7. It is not surprising, therefore, that there is an analogous result to Theorem 3.3.3 for diffeomorphisms on a compact Imanifold without boundary. Topologically, there is only one such connected manifold; namely the circle, S1.
Let Diffl(S1) be the space of orientationpreserving C1diffeomorphisms on S1 with the C'norm. Then Peixoto's theorem for diffeomorphisms on the circle can be stated as follows.
Theorem 3.4.1 (Peixoto) A diffeomorphi f €Diffl(S1) is structurally stable if and only if its nonwandering set consists of finitely many Jxed points or periodic orbits
Figure 3.9 Some examples of structurally stable phase portraits on W2 derived from flows on S2. All fixed points and periodic orbits are assumed hyperbolic. To ensure stability at infinity the corresponding vector fields on S2 have an unstable hyperbolic fixed point at the north pole (cf. Figure 3.4).
3.4 Anosou [email protected] 133
all of which are hyperbolic. Moreover, the structurally stable diffeomorphisms form an open dense subset of Diff '(S1).
Recall (see Proposition 1.5.1) that iff has periodic points then its rotation number p( f ) is rational. Therefore, the structurally stable diffeomorphisms on S1 have rational rotation number (the converse of this statement is not true (see Exercise 3.4.1)). Iff is structurally stable with rotation number y(f) = p/q, in lowest terms, then its dynamics are very simple. It has an even number of periodq cycles with stable and unstable periodic points alternating around the circle (see Figure 1.23).
It was hoped that generalisations of the behaviour described in Theorems 3.3.3 and 3.4.1 would not only characterise structurally stable systems in higher dimensions but would also prove to be generic. To this end, 'MorseSmale' vector fields and diffeomorphisms were defined (Chillingworth, 1976, p. 231 and Nitecki, 1971, p. 88). Unfortunately, it was found that, while such systems were structurally stable, their properties did not characterise structurally stable systems in higher dimensions. In particular, it was shown that there were structurally stable diffeomorphisms on manifolds of dimension n 2 2 (corresponding to vector fields on manifolds of dimension n + I 2 3, by suspension) whose nonwandering sets contained infinitely many periodic points. The Anosov difSeomorphisms of the torus, Tn, are a subset of Diffl(Tn) that exhibit this behaviour. We can describe a diffeomorphism f on Tn in terms of a 'lift' in much the same way as we did for diffeomorphisms of the circle (see 8 1.2.2). In this case, the lift 7 is a diffeomorphism on Wn which satisfies
for each x E a", where n: Rn + T n is given by
n(x) = n((x1,. . ., x,,)~) = (xl mod 1, . . ., xn mod t)T
(see Figure 3.10). If keZn, then (3.4.1) implies
n$(x + k)) = f(n(x + k)) = f(n(x)) = n@(x)), (3.4.3)
for all X E Rn. Continuity of f then gives,
'(x + k) = f(x) + I(k),
where I ( ~ ) E Bn (see (1.2.10) et seque). All lifts of diffeomorphisms on Tn must satisfy (3.4.4).
We will begin by describing a special subset of the Anosov diffeomorphisms that are known as the Anosou automorphisms. The lift of an Anosov automorphism f is a hyperbolic, linear diffeomorphism, A: Rn + Rn, which is such that:
134 3 Structural stability, hyperbolicity and homoclinic points
Together (3.4.5a and b) ensure that A': Z" + Z" and it follows that both A and A' satisfy (3.4.4). Moreover, given that f(n(x)) = n(Ax) and g(n(x)) = n(A 'x), then
f(g(n(x))) = f(n(A 'x)) = n(AA'x) = n(x) (3.4.6)
and
g(f(n(x))) = g(n(Ax)) = x(A 'Ax) = n(x). (3.4.7)
Thus, f': T n , T" exists and has lift A'. Finally, observe that f(x) and f'(x) are differentiable because Ax and A'x are obviously so and n is a local diffeomorphism. Hence f is a diffeomorphism on Tn.
Hyperbolic, linear diffeomorphisms on W n are often referred to as automorphisms because they are isomorphisms of the group W n with itself. The diffeomorphism f defined above is called an automorphism to distinguish it from other Anosov diffeomorphisms whose lift is not linear. A general definition of Anosov diffeo morphisms can be found in Arnold (1973, p. 126) or Nitecki (1971, p. 103) but, thanks to the following result due to Manning (1974), it will be sulficient for us to consider only the automorphisms here.
Theorem 3.4.2 Every Anosov diffeomorphism f of Tn, such that R(f)= T n , is topologically conjugate to some Anosov automorphism o j Tn.
Coupled with Theorem 3.4.2, the following result shows that such Anosov diffeomorphisms have complicated nonwandering sets.
Proposition 3.4.1 A point BET" is a periodic point of the Anosov automorphism I: T n + T n ij and only if B = n(x). where x E 88"has rational coordinates.
Figure 3.10 (a) Angular coordinates on T2 measured in units of 2n. ( b ) The map r identifies each point ( x , , x 2 ) ~ R 2 with a point (01, e 2 ) ~ { ( x 1 , xl)I 0 < xI < 1,0 Q x2 < 1) which in turn defines a unique point on T2.
3.4 Anosov diffeomorphisms 135
Proof. Let 8 be a periodic point of f, then F(B) = 8, for some positive integer q. Suppose x E Rn satisfies n(x) = 8, then
This means that
where x=(x , , . . ., x , ) ~ and m= (m,, . . ., mn)T, i = I , . . ., n. Since A is a hyperbolic matrix, (Aq  I)' exists and (3.4.9) has solution
Now, A*  I is an integer matrix and therefore (Aq  I)' has rational elements. Hence x has rational coordinates.
Conversely, if BET" has representative x = ((plO)/r), . . ., ( ~ : ' ) / r ) ) ~ , where pjO), r E Z with r # 0, then, for any k E Z,
for some integers pp), . . ., pLk'. However, there are at most r" points on T" that can be represented in this way and, therefore, there is a q>O such that n(Aqx) = x(x).
Proposition 3.4.1 not only implies that f has infinitely many periodic points; it also shows that the periodic points are dense in the torus. All these points lie in the nonwandering set R of f and, since R is closed (see Exercise 1.4.2), we conclude that R = T n .
The final piece of the argument against MorseSmale systems was provided by Mather (1967).
Theorem 3.4.3 (Mather) The Anosov diffeomorphisms on T" are structurally stable in Diff'(Tn).
Thus, the Anosov diffeomorphisms on Tn, n 2 2, are examples of structurally stable diffeomorphisms on a compact manifold whose nonwandering set contains infinitely many points. What is more, the dynamics on R is very complicated involving infinitely many periodic orbits densely distributed over the torus. In fact, every periodic point is hyperbolic (see Exercise 3.4.2). Since T n is compact, there can only be finitely many such points with a given period q on the torus. However, periodic points of infinitely many periods can be shown to occur (see Exercise 3.4.3) making up the infinite set predicted by Proposition 3.4.1. To gain some insight into how all this complexity arises let us consider the following well quoted example (see Arnold, 1983; Arnold & Avez, 1968).
136 3 Structural stability, hyperbolicity and homoclinic points
Let A: RZ + R2 be given by
It is easily verified that A satisfies (3.4.5). The behaviour of A on R2 is simple: it has a saddle point at x = 0 with stable and unstable eigenspaces given by the straight lines
1  5112 y = ( ) and y =(I_+;)" (3.4.13)
respectively. The complexity arises when it is mapped down onto T2. Forward iterations of A on RZ have the effect of contracting and expanding along the two perpendicular directions in (3.4.13) as shown in Figure 3.1 l(a). The unit square B , = {(x, j9)IO g x < I , 0 g y < 1) is mapped onto thinner and thinner parallelo grams (Figure 3.11(b) and (c)). The slopes of the longer diagonal of these parallelograms are rational but they approach the irrational $(l t 5'1') for large numbers of iterations. When the points in these images of B, under A are identified with points on T~ (see Figure 3.11(b) and (c)), it can be seen that repeated application off has the eKect of distributing any subset of T 2 more and more evenly over the whole torus. An alternative way of seeing this is to recognise that, for any X E R2, ANx can be made arbitrarily close to the line y = +(I + 51'2)x by taking N to be sufficiently large. Since f(l + 5't2) is irrational, this straight line represents a curve, W", that winds densely around the torus (see Figure 3.12).
The stable eigenspace of the saddle point in Figure 3.1 1(a) also corresponds to a densely wound curve, W" on T2. The key to the complexity of the dynamics o f f lies in the fact that these stable and unstable manifolds intersect in a dense set of transverse homoclinic points (see Figure 3.12). A konloclinic point is one that lies in both the stable and the unstable manifolds of a fixed or periodic point. Such points are said to be transverse if they arise from a transverse, rather than a tangential, intersection of the manifolds. Observe that if flt is a homoclinic point, i.e. Bt E Wsn w", thenf (6 t )~ W% WW" becauseBt E Ws9". Hence f(Bt) is a homoclinic point. Thus the dynamics of these homoclinic points is confined to the dense set of intersection points of W s and W".
The reader must not confuse these homoclinic points with the periodic points off. Recall, by Proposition 3.4.1, periodic points of f have representatives X E R2 with rational coordinates. However, the stable and unstable eigenspaces of A have irrational slope (see (3.4.13)) and, apart from x = 0, no point on them has rational coordinates. We shall.see later (see $3.7) that the occurrence of transverse homoclinic points is indicative of complicated dynamical behaviour.
It is also worth noting that A4 is a lift of P for any q. Thus iff has a periodic point B* of period q, then its stable and unstable manifolds are densely wound curves on T Z parallel to W s and W", respectively. This is because, for each x*
3.4 Anosoo d~J%eomorphisms
Figure 3.1 1 Illustration ofthe toral automorphism given by (3.4.12): (a)
linear saddle point of A = (! f); (b) image A(Bl) of the unit square \ I '/
B, under A; (c) image A2(Bl). The shading in (b) and (c) indicates how n(A(B,)) defines f: T2 +:T2 and ~(A'(B,)) gives f2: T2 + TI.
138 3 Structural stability, hyperbolicity and homoclinic points
satisfying 9* = n(x*), we can write
Aq(x) = Aq(x*) + Aq(x  x*).
Thus, the stable/unstable eigendirections at x* are given by translates of the stable/unstable eigenspaces of A at the origin. It follows that P also has a dense set of transverse homoclinic points for each q. Thus, we can expect the complexity arising from homoclinic points in f, to also occur in fq. This is reminiscent of the 'complexity on all scales' observed in Figure 1.42.
3.5 Horseshoe diffeomorphisms
This is another example of a class of diffeomorphisms which are structurally stable and have a complicated nonwandering set supporting infinitely many periodic orbits. These diffeomorphisms are particularly important because the complexity
Figure 3.12 Segments of the stable and unstable manifolds of the Iixed point, n(O), off on T 2 . Segments AB and A'B' of E" and E' for A have been mapped onto the torus using n(x) = (x mod 1, y mod I) . All inter sections of W' and Wu, except at the origin, are transverse homoclinic points. Note W u ( W s ) always has +ve(ve) slope on 3,.
3.5 Horseshoe diffeomorphisms 139
exhibited by them can be shown to occur in any map that has transverse homoclinic points (see 5 3.7).
3.5.1 The canonical example Consider a diffeomorphism f: Q , WZ, where Q = {(x, y)llxl, 1 yl < 11, that is constructed in the following way. Each point (x, y ) e Q is first mapped to (Sx, y/5) and Q is mapped onto the rectangular region R = {(x, y)llxl< 5, Iyl< 1/51. This region can be divided into fifths by the lines 1x1 = 1 and 3. The map f is completed by bending the central fifth of the rectangle and placing the resulting horseshoe shaped region on the plane in such a way that its second and fourth fifths intersect with Q in Q, and Q, as shown in Figure 3.13(a). Observe that, if Po, P , denote the preimages of Q,, Q,, respectively (i.e, f(Pi) = Qi, i = 0, I ; see Figure 3.13(b)), then fJ P, is linear for i = 0 , l (see Exercise 3.5.1).
We can show that f has a complicated invariant set by considering the sequence of subsets of Q defined inductively by
n e Z t , with Q("=Q0uQ, . It is not difficult to see that Q'z)=f(QouQ,)nQ consists of four horizontal strips lying inside Q 0 u Q , (see Figure 3.14(a)). Moreover, it is apparent that Q'" 3 Q'2' 3 . . . 3 Q(") 2   . and Q'"' consists of 2" horizontal strips (see Figure 3.14(b)). If we consider the intersections of the sets Q(.) with the yaxis then the relationship between the resulting subintervals for successive values of n is easily recognised as a prescription for the construction of
a Cantor set. It follows that the intersection 0 Q'" is a Cartesian product of an n.2 +
interval in x with a Cantor set of yvalues. In a similar way, iterations of the inverse o f f can be used to obtain an analogous set of vertical strips. Some care is needed here because f'IQ is only defined on the subset Q")= Q o u Q , of Q. To avoid this difficulty, we take
and define
for n ~ l i ' . The latter equality in (3.5.3) follows because Q''"'" is a subset of the two vertical strips Q'O' = P o w P , for all n c Z t . Thus, for each n c Zt , the intersection of Q''"'" with Q") = Q 0 u Q , is the same as its intersection with the whole horseshoe f(Q). The map f  ' stretches Q('" '))nQ(l) linearly by a factor of five in the ydirection, contracts it by five in the xdirection and replaces it on the square as shown in Figure 3.15(a,b) for n = 1,2 , respectively. The sets Q'O), Q(", Q'2', . . . then have 2,4,8,. . . vertical strips, respectively (see Figure
3.15(c)) and Q'"' is the Cartesian product of an interval in y with a Cantor nsN
Figure 3.13 (a) Construction off on Q = ABCD, showing rectangular region R divided into fifths labelled from left to right. (b) Preimages Po, P, of Q,, Q,, respectively, are vertical strips consisting of the second and fourth fifths of Q = ABCD.
Figure 3.14 (a) Construction of f(Q,uQ,): stretchingJcontraction yields two strips within the rectangle R; folding yields four horizontal strips for Q(Z)=f(Q(l))nQ. Images of Q, (Q,) at each stage are shown. (b) Illustration of Q'"' for n = l , 2 , 3 . Q'") consists of 2" disjoint horizontal strips whose width, 2/5", rapidly decreases with increasing n; indeed, the eight strips of QC3) (shown in black) are barely resolved in this diagram.
142 3 Structural stability, hyperbolicity and homoclinic points
set of xvalues. If we now define
then A is a Cartesian product of two Cantor sets which is itself a Cantor set.
Proposition 3.5.1 The set A = Qld is invariant under f and f'. "€2
Proof. Let X E A , then XEQ'") for all n ~ i l l , Now if XEQ'"), n~ N, then (3.5.3) implies that f ( x ) s Q(("'))nf(Q) c Q(("'I). I fxeQ(") , n~ Z+, then observe that:
( i ) f (x) E f(Q(")); and ( i i ) ~ ( x ) E f(QiO)) c Q, since x E Q'O).
Figure 3.15 (a) Illustration of Q")= f'(Q'")= PouPI. Notice f' only acts in Qu)=Q0uQ,. (b) Illustration of Q'')=f'(Q(O)nQ(')). The shaded squares in the unprimed part of the diagram represent QIO)n Q"). Q( ' ) consists of the shaded strips in the portion of the diagram labelled with two primes. (c) The vertical strips Q 1 O ) , Q''I, Q(2) , defined by (3.5.3) ate shown. Observe that Q'O)[email protected][email protected]~), . . . . The set @"I
consists of 2"+' disjoint vertical strips each of width .2/5("+ 'I.
C"' Do'
An" B o n C ~ " D ~ "
J replace on Q
3.5 Horseshoe difeomorphisms 143
Together (i) and (ii) imply f ( x ) E f(Q("')n Q = Q("+ ' I . Thus, if x s Q(", for all n E Z, then f ( x ) ~ Q ( " + ') for all ~ E Z . Hence f ( x ) ~ A .
Similar arguments (see Exercise 3.5.2), with the roles of (3.5.1) and (3.5.3)
reversed, show that A is invariant under f' and, therefore, f (A) = A. a The map f , as we have defined it up to now, is not a difleomorphism of the square
( x , Y ) + ( ~ ~ , 5 y ) nn replace on Q
I
Bo' A"' I I
fI I  I I I I
CI1DI' 1 1 1 1 I I I I 1 1 1 1
1 I I I l l 1
Do"CO" BI1'Al"
t replace on L)
t replace on L)
1 44 3 Str~ictural stability, hyperbolicity and homoclinic points
Q (f(Q) $ Q). Moreover, it does not have an obvious connection with diffeo morphisms on compact manifolds without boundary. However, a diffeomorphism, g: S2 + S2 can be constructed such that f is the restriction of g to a subset of the sphere. The first step in the construction is to extend the map f to a capped square Q' as indicated in Figure 3.16. The extension f' is constructed in such a way that f' IF has a unique, attracting, hyperbolic fixed point. This means that once a point is mapped into F its orbit subsequently remains in F. The map f' can, in turn, be extended to a closed disc D 2 of suitably large radius. The extension g': D Z + D 2 is taken to be such that g'(DZ) takes the form shown in Figure 3.17. The diffeomorphism g: SZ + S2 is finally obtained by identifying the disc, D2, on R 2 with a cap. C2, on the sphere (by stereographic projection, see 93.3) and adding a unique, repelling, hyperbolic fixed point in S2\C2.
Since g is a global diffeomorphism on S2, both g and g' are defined for all points of the sphere. However, its construction ensures that g coincides with f o r
Figure 3.16 (a) The capped square Q' = GvQuF; (6) the extension f': Q' + Q' is such that G' = f'(G) and F' = f'(F) are both subsets of F.
Figure 3.17 (a) The disc D2 containing Q'; (b) image of DZ under g' (shaded) with g'lQ' = f'.
3.5 Horseshoe di~eomorphisms 145
f when restricted appropriately. Here we have taken the notational liberty of not distinguishing between these restrictions on the sphere and their representatives on OW2 via stereographic projection. This distinction does not play a significant role in our discussion and, once noted, should not lead to any confusion. The unstable fixed point in S2/C2 means that the ordinary points of this set move towards C2 under g. As we have shown in Figure 3.17(b), g)(D2\Q') is again a contraction and, therefore, g essentially delivers points to Q'. On Q', g behaves in the same way as f'. We already know that f' has an invariant Cantor set A arising from its restriction, f, to Q, but what of the points SZ\A = A'? The following proposition provides part of the answer to this question. It states that those points
of Q' that do not lie on the infinite set of vertical line segments, r) Q("), neN
are eventually swept into F.
Proposition 3.5.2 The orbits under g of points in Q\( C) Q(")) ultimately approach neM
the stable f ixed point of g in F .
Proof. Figure 3.16 shows what flappens to the various parts of Q under a single application of glQ' = f'. The left (L) and right (R) fifths of Q are mapped, together with G and F, into F. Since F contains a unique, attracting fixed point, the orbit of any point in F approaches this point asymptotically. Points in the midfifth, M, of Q suffer the same fate after one more iteration. Such points are mapped into C by f and into F by f2. Only points in P o u P 1 remain in Q (in fact in QouQ,) after one application off'. In other words, points in Q\Q'O' enter F after at most two iterations of g.
Let us focus attention on the partition of the square Q provided by Qt'', rather than Q'O). Observe that points in Q(0)\(Q'O)nQ(l)) are mapped into L, M, R after one iteration of glQ = f (see Figure 3.15(b)) and thence into F after two or three iterations. Thus we conclude that all points in (Q\Q(0))~(Q(O)\(Q(O)nQ(l))) = Q\(Q(O)nQ(I)) enter F after at most three applications of g. Similarly, if we consider the partition of Q provided by Q'2), we mnclude that points in Q( 1 ) \ ~ (  2 ) = (Q(o)).Q(~) ) \ ( Q ( ~ ) ~ Q (  ' ) ~ Q (  ~ ) ) have images under g in
1
Q(O'\(Q(O)nQ(')). Therefore, if XEQ\ C) Q(") then g"(x)~; F for k > 4. Thus, we n = O
conclude inductively that all points in Q\ C) Q("' ultimately enter F. n s N
0
In view of the construction of A, it is clear that A E 0 Q'"'. Moreover, it is not neN
difficult to show that 0 Qt"' is invariant under g (see Exercise 3.5.3). Bearing REN
146 3 Structural stability, hyperbolicity and homoclinic points
in mind that r\ Q(") is a set of straight line segments parallel to the yaxis and . . neN
that g involves a contraction along that direction, it is not surprising that points
in (0 Q("))\A have orbits that approach A asymptotically (see Exercise 3.5.3). n s N
It must be emphasised that these orbits are not confined to a single vertical line
n s N
Let us now turn to the dynamics of points in SZ\Q'. Unlike f', g' is defined for all x E Q (see Figure 3.18). Of course, g ' IQ") z f ' is as illustrated in Figure 3.15 but g'(Q\Q(l)) c S2\Q' = Q'E. This means that points in Q\QU) are the images under g of points lying outside Q'. We have already discussed the fate of such images under forward iterations of g. Since An(Q\Qtl))= @, points in
(Q\Q( ' ) )n(n Q(')) have orbits approaching A asymptotically; while those in neN
(Q\Q('))\ n Q(" have orbits entering F. However, under revers,e iterations of g n s N
Figure 3.18 Illustration of g ' showing that points starting in Q\Q"' = T u M H u B are mapped out of Q' by g'. Observe that SZ\Q' contains a unique, stable, hyperbolic fixed point of g' so that the orbits of these points under g' do not return to Q'.
3.5 Horseshoe diffeomorphism. 147
we are able to extend the invariant set 0 Q(") onto the whole sphere. The ncN
resulting set of points is called the inset, in(A) of A, i.e.
Similar arguments to those presented above, with g' replacing g, lead to
analogous conclusions about the infinite set of horizontal lines 0 Q(#) (see naZ +
Exercise 3.5.3). It follows that there is a set of points, out(A) s G u Q u F, the outset of A, whose orbits approach A under reverse iterations of g, i.e.
out(A) = { x ~ S ~ ( g  ~ ( x ) + A as nr a). (3.5.6)
The role of the set A in the dynamics of g is clearly analogous to that of a saddle point in simpler diffeomorphisms. The inset and outset of A generalise the stable and unstable manifolds of the saddle. We will return to sets possessing this more general hyperbolic structure in $3.6 but now we must consider the dynamics of g on A.
3.5.2 Dynamics ON symbol sequences Let C be the set of all biinfinite sequences of the binary symbols {O,l), i.e. C = {ula: Z , {O, 1)). The elements, a, of C are called symbol sequences and they are defined by specifying u(n)= a n € {0,1) for each n E Z. We will write u = {a,}:= _, = {. . . 626 160.U162 . . .). Our aim is to study the dynamics of the map a: C + T: defined by
a(a),=a,,, (3.5.7)
n e Z . This is known as a ldtshift on C because it corresponds to moving the binary point one symbol to the left.
Proposition 3.5.3 The ldt shift a: C , C has periodic orbits of all periods as well as aperiodic orbits.
A point a*eI: is periodic if
aq(a*) =a*, (3.5.8)
q~ Z +. If q is the least, positive integer for which (3.5.8) is satisfied then a* is said to be of period q. It is not difficult to see that (3.5.8) will be satisfied if and only if a,+ = a,*,,, for all n~ Z. It is then easy to find periodic points of a with any given period, q. The required sequence, a*, is generated by repetition of a block of symbols of length q that is itself not composed of repetitions of any of its subblocks. For example, the point
148 3 Structural stability, hyperbolicity and homoclinic points
has period1 4, while 
a* = {. . . 10101 1010 101 1010~101 . . .) (3.5.10)
satisfies aI4(a*) = a* but has pen'od7 because a7(a*) = a*, also. It is eqt,ally straightforward to show that a has aperiodic orbits. For instance,
which contains symbol blocks of the type shown for all n e Z + , is such that there is no q E E * such that @(a) = a.
Proposition 35.4 There is a topology in which the periodic points of cc are dense in C.
There is a natural way of defining how close two symbol sequences are to one another. Given two sequences in C , we can obtain the length of the largest symbol block, centred on the binary point, on which they agree. The larger the size of this block the closer the two sequences are deemed to be. We are then able to define the limit of a sequence of elements in C. A sequence {a(")},"=, G C is said to tend to U E C as m + oo, if, given NEE', there exists M EE' such that a?) = an for (N  1) ,< n < N, when m > M. Clearly, if a("'+ o as m + oo then a'") and a agree on increasingly large central blocks. For example, the sequence d'"defined by
converges, as m+ oo, to the sequence a with a, = 1 for all ~ E Z . With the above definition of convergence, periodic points of a are dense in C.
This follows because, given any UEC, there is a sequence of periodic sequences {a(")},"=, which tends to a as m + w. Each sequence a'") is simply taken to be periodic with period 2m and such that a',") = a n for  (m  1) < n < m. As an example, let a be the aperiodic sequence (3.5.1 I), for which
 a(l) = . . . 01010~10101 . . . ,
 a(2' = . . .I010 10.10 1010.. . ,  (3.5.13) d3'= . . .010101010.101010101.. . , d4' = . . .10101011 1010.101 1 10101011.. .
and so on.
Proposition 33.5 The left shift a: Z + Z has a dense orbit on Z.
To justify Proposition 3.5.5 we must show that a has an orbit on C that approaches every point of C arbitrarily closely. Let U E Z be such that a_ , for n~ N is given by
3.5 Horseshoe diffeomorphisms
the following ordered lists of symbol blocks:
(i) all blocks of length 1, i.e. {0), (1); (ii) all blocks of length 2, i.e. {O,O), (0, 11, {I, 01, (1, I); (ii) all blocks of length 3; and so on.
All possible blocks of all lengths are included in {a_,},"=,; a,, n E Z+, can be chosen arbitrarily. The orbit of a under a contains {am(a)lrn~ N}. Now, by construction a contains any given symbol block of length N in its left hand half. After sufficiently many applications of a this block will be centrally placed about the binary point. Since N is arbitrary, any element of C can be approximated arbitrarily closely by some point on the orbit of a under a.
In view of the rather special construction used above to obtain a sequence a whose orbit under a is dense in C, the reader may feel that such sequences are in some sense rare or atypical. This is not the case. In fact, most binary biinfinite sequences contain any prescribed block of symbols (see Hardy, 1979) and therefore have a dense orbit under a. The particular example chosen above is carefully ordered purely to make the argument more convincing.
Having established some properties of the left shift a: Z + C, we must reveal our motive for examining the dynamics of this map: namely to obtain a symbolic description of the dynamics of the horseshoe diffeomorphism on A. Before doing this, it is worth noting that the validity of Propositions 3.5.35 does not depend upon the binary nature of the sequences in C. Similar results can be derived for sequences of msymbols, (0, I , . . ., m  1 ) say (see Exercise 3.5.5). Binary symbol sequences allow us to deal with the horseshoe map of $3.5.1. However, there are more sophisticated maps of this type (see Exercise 3.6.5) whose 'symbolic dynamics' involve sequences of m symbols with m > 2 .
3.5.3 Symbolic dynamics for the horseshoe difPeomorphism In this section we show that the restriction of the horseshoe diffeomorphism to the invariant set A is topologically conjugate to the left shift a on 2. The key idea is that the points of A can be 'coded' as biinfinite sequences of (0, 1).
Recall that A = n Q'"), where Q'"', n e Z + , is the disjoint union of 2nhorizontal nrZ
strips on the square Q, while Q'"', nEN, is the union of 2"" similar vertical strips. As Figures 3.14(b) and 3.15(c) illustrate these sets of strips are 'nested', i,e. Q(') 2 Q(2) 2 . . . 2 Q(,) 2 . . . and Q'O) 2 Q(') 2 . . . 3 Q(") 3 . . ..Thus, A(") =
N
r / Q(n)=Q(wo)nQ(N) is the disjoint union of 2'" squares of side 215" n = (N 1)
(see Figure 3.19). Clearly, as N + w, the size of the squares tends to zero, their number becomes infinite and A'N) + A. The coding of the points of A follows from the fact that each square of A(N' can be uniquely represented by a symbol block, dN)= {a(N1,. . . IJ,,.~~. . .aN}, u,E{O, 11, of length 2N.
Any given strip in Q'") can be allocated either 0 or 1 in the following way.
150 3 Structural stability, hyperbolicity and homoclinic points
Consider the vertical strips Po and PI. Observe that
Q"'= g(Po)ug(Pl)=QouQ~ (3.5.14)
where QonQ1 = 0 (see Figure 3.13). Furthermore,
Q"' C_ gZ(Po)ug2(Pi)
with g2(Po)ng2(Pl)= 0 (see Figure 3.14(a)). In general, for n e Z + ,
Q'"'s Ei"'Po)ugn(P,) (3.5.16)
and g"(P0)ngn(P,) is always empty because P o n P l = 0 and g is a diffeo morphism. Thus a horizontal strip of Q(") lies either in g"(Po) or f(P,). We allocate the symbol 0 to a strip of Q'") if it is a subset of g(Po) and the symbol 1 if it lies in gn(P,) (see Figure 3.20). Obviously, these symbols alone do not provide a unique description of each horizontal strip in Q(") for n 2 2, however, they can be used to obtain one. For example, two strips of Q(2) have been allocated the symbol 0 but they are distinguished by the k t that one lies in g(Po) (i.e. strip 0 of Q(") and the other lies in g(P,) (i.e. strip 1 of Q'')). Hence the strips in Q") can be
Figure 3.19 Illustration of   (N  I t Q(") for (a) N = I; (b) N = 2. The square regions defined by (3.5.17) with d2) given by (11.01) and {lo. 11) are indicated.
3.5 Horseshoe diffeomorphisms 151
uniquely labelled by giving two symbols: the first specifying a strip in Q") so that the second uniquely determines a strip in QC2) (see Figure 3.20). Similarly, the strips of Q'j' can be uniquely labelled by starting from the unique labelling of the strips in Q"' and appending the symbols allocated to QO). It follows that the strips of Q'" are uniquely specified by a set of n of the symbols (0, 1). Similar arguments can be carried through for Q(") by considering the images of Po and P I under powers of g' (see Exercise 3.5.8). A vertical strip of Q'"), neN , is allocated the symbol i if it is a subset of g"(Pi), i = 0 , l . For n ~ H+, unique labels for the strips of Q("' are obtained by appending these allocated symbols to those of the strips in Q((nl)) (see Figure 3.21). Notice we have appended symbols on the left so that the order in the strip label matches that of the negative integers. Finally, we
Figure 3.20 Coding of strips in Q'"' for (a) n = 1 ; (b) n = 2; (c ) n = 3. The symbol allocated to each strip is shown on the left and the unique code for the strip is given on the right.
152 3 Structural stability, hyperbolicity and homoclinic points
can construct the symbol blocks representing the squares occurring in A'"'. Each such square is the intersection of one of the vertical strips of Q''Nl)) with one of the horizontal strips of Q". If the vertical strip has label u(N ,), . . ., 00
and the horizontal strip has label u,, . . ., aN, the symbol block representing the square is taken to be at")= {a_,, ,,, . . . ,a  , , ao .a l , . . ., a,}. Thus, for example {11.01) and (10 . l l ) , respectively, represent the top right hand and bottom left hand squares in the illustration of A") given in Figure 3.19(b). It is not difficult to show (see Exercise 3.5.11) that the square represented by the symbol block atN'
Figure 3.21 Coding for strips of Q'") for n =0, 1, 2: (a) Q'O'; ( b ) Q'"; (c) Q(2). Unique labels for the strips are given above and allocated symbols below. Notice that, to match the negative integers, symbols are appended to the left rather than to the right.
3.5 Horseshoe dtfleomorphisms 153
is given by
In the limit N + co, the above construction assigns a unique, biinfinite binary sequerice with each point of A. Moreover, (3.5.17) allows any such sequence to be converted to a unique point of A. We have therefore constructed a bijection h: C + A .
Proposition 3.5.6 The bijection h: C + A dejined above is a homeomorphism that exhibits the topological conjugacy of g: A + A and a: Z + C.
Proof. The nested nature of vertical and horizontal strips defining A means that sequences that are close, in the sense that they agree over large central blocks, map under h to points of A that are geometrically close together. Similarly if two points of A are geometrically close, the symbol sequences agree over a large central block because it is only for N sufficiently large that such points are distinguished in A'"'. Thus h is a homeomorphism.
Let ~ E C and
Then
= h(a(a)).
Therefore h exhibits the conjugacy of g and a .
Proposition 3.5.6 implies that the complexity exhibited by the orbits of points in Z under a (see $3.5.2) also occurs in the orbits of points of A under g. Thus g(A has infinitely many periodic points, its periodic points are dense in A and it has orbits that are themselves dense subsets of A.
Another feature of the dynamics of a : Z + C, that has important repercussions for gJA, is that there are points in C whose orbit under a is aperiodic. Since a(a) and a are not, in general, close in C, these orbits wander throughout Z in an apparently disorganised way. Similarly, their counterpart in the orbits of glA move around A by hopping from point to point in a random or chaotic way. Indeed,
154 3 Structural stability, hyperbolicity and homoclinic points
invariant sets like A are often referred to as chaotic sets (see $3.6) because of the presence of such orbits.
While the dynamics of glA is very complicated, we must not forget that the dynamics of gJAc suggests that A is, in some sense, hyperbolic. In the following section, we consider how such sets fit into a general theoretical framework.
3.6 Hyperbolic structure and basic sets
Let us return to the hyperbolic nature of the invariant set A of the horseshoe diffeomorphism g: S2 + SZ. In fact, A is said to have a hyperbolic structure or to be a hyperbolic set for g. Our aim in this section is to explain this statement and to introduce an important theorem about diffeomorphisms whose nonwandering set, R, has a hyperbolic structure.
It is helpful to review our previous encounters with hyperbolicity (see 92.1 and 2.2). The striking feature is that, thus far, we have only had to consider hyperbolic fixed points. Nontrivial hyperbolic sets such as a hyperbolic periodic orbit, or a normally hyperbolic invariant circle, are defined in terms of a hyperbolic fixed point of a related map (fq or 9 in 92.2). We are then involved with the local behaviour of a map at a fixed point in a Euclidean or Banach space. In such cases, the hyperbolic nature of the fixed point is given in terms of the eigenvalues of the derivative map (DP or D f ). It is not possible to use this approach to characterise the hyperbolicity of the invariant set, A, of the horseshoe diffeomorphism. However, it is useful to consider why such an approach fails. There are two problem areas.
(i) The horseshoe diffeomorphism is defined on a manifold (the sphere) and not a Euclidean space. This means that the generalisation of the derivative map to this situation must be considered.
(ii) The complexity of A is such that it is not possible to formulate the problem in terms of a fixed point of some related map. For example, A contains aperiodic orbits which do not correspond to a fixed point of g4, for any q~ Z'. Thus, having introduced the appropriate generalisation of the derivative map, our definition of hyperbolicity must allow for the fact that x and g(x) are different points in A.
Let us begin by considering how the results of $2.2 can be applied to a diffeomorphism f: M t M when M is an ndimensional, differentiable manifold that is not a subset of R" (see Figure 3.22). The derivative map, Df(x*): Rn + Rn, used to discuss the hyperbolic fixed point, x*, of f: Rn + Rn in 92.2, is replaced by the tangent map Tf,.: TM,. + TM,., where TM,. is the tangent space to M at x*. Recall (see $ 1.3) that TM, can be defined, for any x e M, in terms of equivalence classes of curves on M with the same tangent vector at x. To see the connection with the behaviour of f near x*, let q(t), with t ~ l ~ R, OEI and q(0) = x*, be a parametrised curve on M passing through x*. To find the tangent vector at x*, we need to differentiate q(t) with respect to t and (see $ 1.1) this can only be done by using a local chart, (U,, ha) say, containing x*. The local representatives, i, $ and G, of f, q and fq, respectively, in (U,, ha) (or, more
3.6 Hyperbolic structure and basic sets 155
concisely, arepresentatives) are given by
They satisfy the equation
(fG),(t) = j,(ija(t))
which, provided M is a C1manifold, can be differentiated to give
(f%,(o) = ~i~(ij ,(o))$fl (3.6.3) A 2
at t = 0. The vectors (f..rl,)(O) and &(0) are arepresentatives of elements of TM,. in (U,, ha). Strictly speaking, they lie in the tangent space to U, at 2: = ij,(O) but, as TU,; is a replica of Wn, this distinction is not always apparent. The derivative map ~'C,(lt) is the local representative of the tangent map Tf,.. As the opening remarks to $2.2 suggest, x * ~ M is said to be a hyperbolic fixed point off: M t M if 2: is a hyperbolic fixed point of in the sense of Definition 2.2.1, i.e. if D?,(z:) has no eigenvalue with unit modulus. What is more, if we assign a metric to TUG then (see Exercise 2.1.2) hyperbolicity of ii: corresponds to imposing bounds on (D<(Z:)"V( for all v in the stable and unstable eigenspaces of D],(E:). Of course, it is only by making such an assignment that we can define a metric on TM,..
Figure 3.22 For a hyperbolic periodic orbit, the Invariant Manifold Theorem gives the existence of stable and unstable manifolds in each chart for T. The charting homeomorphisms allow them to be transferred to M (shown here as S2). E";: are tangent to the images of Wz," on any local chart.
156 3 Structural stability, hyperbolicity and homoclinlc points
Indeed, if x* lies in the overlap of two charts (U,, ha) and (Up, hp), then we are only allowed to choose metrics on TUG and TU,; that, for all V E TM,., satisfy Ii,I,= IS,q(b, where i, (ip) is the a (P)representative of v (see Figure 3.23). The common value defines Ilvll,. for any v in TM,.. Compatible metrics, such that IlvIl, is positive definite, defined at all points x of all overlaps of an atlas provides a Riemannian structure for M (see Exercise 3.6.2). If M is equipped with a Riemannian structure then we can express the hyperbolic nature of x* in a coordinatefree way by requiring that:
(i) TM,. = F,. $ E$, where E$" is the stable (unstable) eigenspace of Tf,.; (ii) there exist c , C > 0 and 0 < p < 1 such that, for every n~ Z',
(((x*)"(v)llxr < Cpnllvllx* for all VEE:., (3.6.4)
II(Tf,.)"(v)ll,. > c,u"llvll,. for all v EE:.. (3.6.5)
With this in mind, let us consider an alternative definition of a hyperbolic periodic orbit.
Let I: M + M have a qperiodic orbit, A(q' = {x,, x,, . . ., x, ,). Of course, each point x, = f'(x,) is a fixed point of P, but we will resist the temptation to use this to test for the hyperbolicity of Afq). Instead, let us use the approach discussed above. The new feature in this case is that Tf,, maps TM,, to TM,,,,,, where f(x,) # x,. However, the Riemann structure on M allows us to deal with this change because it provides a norm, 11. II,, for every TM,, x E M. We must therefore
Figure 3.23 Schematic representation of the definition of a norm, (1 II,., on TM,. in terms of compatible norms 1.1, and ( . I b on TU,: and TU,), respectively.
3.6 Hyperbolic structure and basic sets 157
generalise (3.6.1)(3.6.5) as follows. Let x, and f(x,) belong to charts (U,, ha) and (Up, ha), respectively, then
w
tp=hp.f.h, ' , ij,=h;q, (fq)=hs(f.4) (3.6.6)
with
Differentiating with respect to t and setting t = 0 gives &
ip = (f.q)(O) = D&(~~,(o))$,(o) = ~?,s(%,)t. (3.6.8) &
Now, i, = $,(o) is a arepresentative of V E TM,, whilst ip = feq(0) is a P representative of TC,,(V)E TM ,,,,,. Thus, the tangent map Tf,,: TM,, + TM ,,,,, and the familiar eigenspace decomposition of TM,, can no longer be used. Instead we require that, for each x,, there exist subspaces F,, and E:, such that TM,, = E",, @ E:, and Tf,,(F;,") = Q(".,,. Of course, the existence of such a decomposition is assured if x, is a hyperbolic fixed point of fq. Finally, it is important to remember that the appropriate norms must be used in the generalisations of (3.6.4) and (3.6.5), i.e.
The above discussion motivates the following definition of hyperbolic structure for more general invariant sets.
Definition 3.6.1 An invariant set A is said to be hyperbolic for f (or to have a hyperbolic structure) i f f o r each x EA the tangent space TM, splits into two linear subspaces E, E: such that:
(i) T f , ( F ) = Eii",,; (ii) (3.6.9) and (3.6.10), with x , ~ x , are satisfied for all positive integers n;
(iii) the subspaces E",, E: depend continuously on x e A.
Item (iii) is trivially satisfied if A is a periodic orbit, since the points X E A are isolated. However, it is an important technical restriction for invariant sets containing a dense orbit or a dense subset of periodic orbits.
It is not dificult to accept that the invariant set A = n Q'"' is a hyperbolic set neZ
for the horseshoe diffeomorphism g. Observe (see 43.5.1) that the set of vertlcal
line segments, n Q("', on the square Q give rise to curves on S2 analogous to neN
the stable manifold of a periodic orbit. Similarly, the horizontal line segments,
n Q("), lead to the analogue of the unstable manifold of the periodic orbit. At n€Z+
each point XEA we can identify tangents to these curves to obtain E", and E:. Moreover, this splitting into F, and E," depends continuously on x, because, for
158 3 Structural stability, hyperbolicity and homoclinic points
any x, x'EA, 4 and E:., (or E: and E:.) are tangent to diffeomorphic images of parallel line segments on the square Q. Finally, the contraction on E: and expansion on E: satisfy (3.6.9) and (3.6.10) withp > i a n d c = C = 1, so that A hasa hyperbolic structure.
In 8 1.4, we noted that fixed points and periodic orbits are invariant sets that frequently appear to attract or repel the orbits of points not contained in them. What is more, they are rather special in so far as they have no proper subsets that are themselves invariant. The following theorem for diffeomorphisms whose nonwandering set, R, has a hyperbolic structure, provides the theoretical basis for these observations.
Theorem 3.6.1 Let f : M + M be a diffeomorphism on a compact manifold without boundary with a hyperbolic nonwandering set R. If the periodic points of f are dense in R, then R can be written as a disjoint union of flnitely many basic sets R,, i.e.
Each Ri is closed, invariant and contains a dense orbit of f. Moreover, the splitting of R into basic sets is unique and M can be decomposed as a disjoint union
where
is the inset of 4.
Diffeomorphisms with hyperbolic nonwandering set, R, and periodic orbits dense in Rare usually referred to as axiomA diffeomorphisms (see Chillingworth, 1976, p. 240; Nitecki, 1971, p. 189). Clearly, any diffeomorphism whose nonwandering set consists of a finite number of fixed points or periodic orbits is axiomA. Moreover, fixed points and periodic orbits are closed, invariant sets that trivially contain a dense orbit, i.e. they are basic sets.
Theorem 3.6.1 does not merely give a decomposition of R. Equation (3.6.12) states that every X E M belongs to the inset (or equivalently, the outset) of one and only one basic set. This means that the wandering points move between the basic sets approaching those that are attracting asymptotically. Some simple examples are illustrated in Figure 3.24.
The horseshoe diffeomorphism, g, on the sphere is a more substantial example. The nonwandering set, R, of this diffeomorphism consists of the invariant set
A = r) Q'") and the two fixed points; one stable and one unstable. R has a neZ
hyperbolic structure and Proposition 3.5.4 shows that the periodic points of g are dense in 0, so Theorem 3.6.1 applies. Obviously, each fixed point is a basic set
3.6 Hyperbolic structure and basic sets 159
(R, unstable; R, stable, say) but can A be decomposed into a number of basic sets? Propositions 3.5.5 and 3.5.6 show that A contains a dense orbit of g. This means that further decomposition of A is out of the question and, since it is also closed (it is a Cantor set) and invariant, the only remaining basic set (R,) is A itself. Basic sets of this type are referred to as chaotic sets (see 93.5.3 and Exercise 3.5.1 1).
The Anosov automorphisms provide another illustration of Theorem 3.6.1 involving a chaotic basic set. Recall that the periodic points of these maps are dense in T" and the nonwandering set is the whole torus. Moreover, in the
twodimensional example with A = ( :), discussed in $3.4, it is clear that F,,
and E",, are given by (1, (1  5'/')/2) and (1, (1 + 5'12)/2), respectively, at every periodic point x, (see Exercise 3.4.2). Continuity requires that this be so for each x E R because the periodic points are dense. The splitting of the tangent space is
Figure 3.24 Illustrations of Theorem 3.6.1 where the basic sets are fixed points and periodic orbits. (a) f: S2 +S2 has nonwandering set, R, consisting of two basic sets, Q, and Q2  both fixed points. All wandering points have alimit set R, and olimit set R,. (6) The basic sets R, and R2 are unstable fixed points, R, is a saddlelike 4cycle and Q, is a stable 4cycle. The dynamics of the wandering points are shown schematically on the right. (c) The basic sets are ail fixed points in this case: R, is unstable; R2 is stable and R,, R, are saddlelike. Once again a schematic representation of the dynamics of the wandering points is given.
._ .. 0..
O . ..0..
__. __. On,
160 3 Structural stability, hyperbolicity and homoclinic points
therefore trivially continuous, being the same at every point of Q. Hyperbolic rates of contraction and expansion follow from the hyperbolicity of A (see Exercise 3.6.4). Hence Cl has a hyperbolic structure. In this case, there is only a single basic set Q, = R = T2 and it follows from Theorem 3.6.1 that the toral automorphism must have an orbit which is dense in the torus.
A further example is the transformation, f, of the solid torus, T = S1 x D2, shown in Figure 3.25. The torus is treated rather like a solid rubber ring. It is stretched (with consequent loss of crosssectional area), twisted and folded to fit inside itself. Repeated application of this transformation results in longer and longer tori, wrapped around T increasingly many times. If the disc D2 is a crosssection of T, . 
is a Cantor set. Thus, the culimit set of f is locally the
product of a Cantor set and a onemanifold. This example has the important property that the chaotic basic set is an attractor.
The set A = n Q(") in the horseshoe diffeomorphism has only a onedimensional n d
inset. This means that most orbits are not asymptotic to A and this makes A
Figure 3.25 (a) Illustration of a transformation f of the solid torus T which has an attracting chaotic set. The image, f(T), of T under f is shown shaded. (b) Intersections of successive images of T under f with a cross section D~ of the torus. Notice that f ' is not defined on the whole torus, however, only forward iterations are required to observe the attracting set. The mapping f is sometimes called the 'spinning difieomorphism'.
(a)
3.6 Hyperbolic structure and basic sets 161
difficult to observe in numerical experiments. In principle, we can find points whose orbits under g remain in a given neighbourhood of A for an arbitrary number of iterations. However, in practice most plotted orbits spend at most a few iterations near A before vanishing into the sink in F. This is because, with finite computer arithmetic, we are unable to approximate in(A) closely enough to sustain a presence near A in the face of repeated fivefold expansion (see Exercise 3.5.4). Thus naive computer experiments involving the orbits of wandering points d o not reveal much about the position of A let alone the dynamics on it. Some feeling for the latter aspect of g can be obtained by using symbolic dynamics (see Exercise 3.5.1 1).
In view of these practical difficulties, it is not surprising that chaotic basic sets are much easier to observe in numerical experiments if they are attracting. Attracting chaotic sets  often referred to as strange attractors  have been observed in a wide variety of computer experiments (see Figures 3.263.30). Detailed docu mentation of this area can be found in Gumowski & Mira, 1980; Helleman, 1980; Lichtenberg & Lieberman, 1982; Sparrow, 1982). Such attracting sets are not fully understood and may not be basic sets in the sense of Theorem 3.6.1. However, they d o appear to have the common property that points in them occur on finer and finer scales. For example, in Figure 3.27, the Henon attractor appears to be onedimensional and to consist of a number of segments. Closer examination reveals that each 'segment' consists of several closely spaced curves of similar shape
Figure 3.26 The Dufing attractor (see Guckenheimer & Holmes, 1983, pp. 8291 & 1913). The Dulling equation can be written in the form
This system is periodic in 0 and the phase space can be taken as M = W2 x S1. Every surface O=constant is a global Poincare section so that the system behaviour is completely described by the PoincarC map P,,. Numerical approximations to P , , appear to have a chaotic attracting set  the Euler approximation is shown in this diagram for &a = 0.4 and cb = 0.25. The structure of P,, is discussed in greater detail in $3.8.
3 Structural stability, hyperbolicity and hornoc,linic points
Figure 3.27 The HCnon attractor (see Hbnon, 1976). The map f producing this attractor is defined by
( x , y ) A ( y  ax2 + I, bx),
where a, b f R . The attracting set for a = 1.4 and b =0.3 is shown. It arises from the repeated folding and stretching brought about by the action off. When magnified the attractor is found to consist of many curves, of similar shape to those resolved above, occurring very close together. This 'braided' nature of the attractor appears to be repeated on all scales.
Figure 3.28 Guckenheimer et al. (1977) discussed a Leslie model of a density dependent population with two age classes of size x and y. It is a discretetime model and the dynamics of the two classes are represented by the map
where r is a real parameter. The map appears to exhibit chaotic behaviour for r 2 17 and a typical orbit for r = 20 is shown here. More numerical details can be found in Guckenheimer et a!. (1977), where the origin of the attracting set is discussed in terms of a twisted horseshoe map.
3.6 Hyperbolic structure and basic sets
Figure 3.29 The Lorenz attractor (Lorenz, 1963). It must be pointed out that there is a theorem corresponding to Theorem 3.6.1 for flows so that strange attracting sets can also arise in flows that are not the suspension of a dilfeomorphism. The Lorenz equations
i = 10(y  x), jl = x(28  z )  y, i = xy  (813)~.
have fixed points at (+6(2'12), +6(211'), 27). The system does not have a global section so the projection onto the xzplane is shown. The orbit generated by using the Euler method with step length of 0.005 and initial point (x, y, z) = (0 .1 ,0,0) is plotted. The projected orbit switches between revolving about (x , 2 ) = (+6(2'j2), 27) and (x, I)= (6(2'12), 27) in an apparently random way.
Figure 3.30 The RBssler attractor (Riissler, 1979). This is another threedimensional flow exhibiting an attracting set with complex dynamics. The system equations are
i =  ( y f z ) , j = x + e y , i = f + x z  p z .
A perspective view of an orbit near to the attractor is shown fore = 0.17, f =0.4 and p = 8.5. It is obtained by using the Euler method with step length 0.005 to approximate the trajectory through (x, y, z ) = (1 ,0 ,0 ) and plotting u = x + y , v = y + z .
1 64 3 Structural stability, hyperbolicity and homoclinic points
that are not resolved on the scale of Figure 3.27. Further magnification shows that each of the latter 'curves' has a similar structure, and so on. The attractor is said to have a 'braided' nature which is repeated on all scales. The reader will recall that the chaotic basic set A of the horseshoe diffeomorphism has this property (see Exercise 3.5.9). In fact, a theoretical connection with the basic set of the horseshoe diffeomorphism can be made in some cases: namely when homoclinic points occur.
3.7 Homoclinic points
We have seen that homoclinic points occur in the dynamics of Anosov auto morphisms. They also occur in the horseshoe map. Consider for example the fixed point represented by the sequence {. . .1111~1111.. . .). The stable manifold of this point on Q is a vertical line segment and the unstable manifold is a horizontal line segment. The effect of a single iteration of the horseshoe map f of $3.5.1 is shown in Figure 3.31. Clearly, transverse homoclinic points must occur. Are homoclinic points a feature of chaotic basic sets? The following theorem provides a partial answer to this question.
Let M be a compact twomanifold and Diffl(M) be the set of all CL diffeomorphisms on M. The elements of a residual subset of DiB1(M) have the property that all their fixed and periodic points are hyperbolic and all intersections of stable and unstable manifolds are transverse. Diffeomorphisms in this subset are usually referred to as KupkaSmale difleomorphisms (see Chillingworth, 1976; p. 227; Nitecki, 1971, p. 83).
Figure 3.31 A transverse homoclinic point xt of the fixed point x* = {. . . 11.11 . . .} of the horseshoe map.
3.7 Homoclinic points 165
Theorem 3.7.1 (SmaleBirkhoff) Let f€DiB1(M) be KupkaSmale and xt be a transverse homoclinic point of a periodic point x* of f. Then there is a closed subset A of B(f), containing xt, such that:
(i) A is a Cantor set; (ii) P(A) = A for some p E Z ' ; (iii) fP restricted to A is topologically conjugate to a shift on two symbols.
A point xt is a homoclinic point of a periodic point x* of period q if it lies a t an intersection (fx*) of the stable and unstable manifolds of the fixed point of P at x*.
The idea behind the proof of Theorem 3.7.1 is illustrated in Figure 3.32. If the stable and unstable manifolds of the hyperbolic saddle point x* intersect at some point x f , then they must intersect infinitely many times. Recall (see $3.4) if xf E W k Wu then P ( x f ) ~ W% Wu for every mEZ. Figure 3.32 illustrates the effect this constraint has on the two manifolds if we attempt to return them directly to x* itself. As the unstable manifold approaches the saddle point the loops between adjacent homoclinic points are stretched parallel to W k and squeezed parallel to Wiw. The manifold therefore undergoes oscillations of increasing amplitude and decreasing period. The fate of the stable manifold is similar under reverse iterations resulting in the homoclinic tangle shown in Figure 3.32.
The connection with shifts on twosymbol sequences is apparent if we consider the images of a small 'parallelogram' R, containing xf and with sides parallel to W%nd Wu, under forward and reverse iterations. For m > 0, the mth iteration of f stretches f'""(R) along Wu and contracts it along Wf Remember, ffm"(xf) is a homoclinic point and belongs to f'""(R) for every m. Eventually, for some N E Zt , f N ( ~ ) , takes the horseshoe shape R, (see Figure 3.32). For reverse iterations
Figure 3.32 illustration of the homoclinic tangle occurring at a hyperbolic saddle point. The parallelogram R has images R, = fN(R) and R, = /  w ( R ) intersecting in a horseshoe configuration.
166 3 Structural stability, hyperbolicity and homoclinic points
the roles of Ws and W u are reversed and, for some N'E E +, fiN')(R) = R,, where R, and Ro intersect as shown in Figure 3.32. Clearly, if p = N + N', fP(Ro) = R, and we would expect fP to exhibit horseshoelike behaviour, i.e. be conjugate to a left shift on twosymbols. The homoclinic point referred to in Theorem 3.7.1 would in this case be xt = fN'(xf).
Theorem 3.7.1 means that f exhibits all the complexity of the left shift a: Z +I: discussed in $3.5.2. In particular, in every neighbourhood of a transverse homoclinic point o f f , there is a periodic point. By Theorem 3.7.1, the transverse homoclinic point xt E A and fP(A is topologically conjugate to the left shift a: X , C. However, by Proposition 3.5.4 the periodic points of a are dense in C. Hence, periodic points of fPlA are dense in A and, therefore, there is a periodic point off arbitrarily close to xt. Thus there are infinitely many periodic points in any neighbourhood of xt.
It is important to realise that Theorem 3.7.1 employs sulficient conditions to ensure the existence of A. As Smale has pointed out (see Smale, 1963), we might expect a similar result to hold with weaker constraints on f. Figure 3.32 suggests that the key requirement is that the stable and unstable manifolds of a hyperbolic fixed or periodic point intersect transversely. With this in mind, the following example shows that the remarkable phenomena described above really do occur. Let us examine the planar map
XI =X+Yl, Y, = y + kx(x  I), (3.7.1)
numerically, for 0 < k < 4. This map has fixed points at (x, y) = (0,O) and (1,O) for all values of k. The fixed point at (0,O) is nonhyperbolic. The linear approximation to (3.7.l)at (0,O)isconjugate to an anticlockwise rotation through angle 8, where
2 sin 6 = [k(4  k)I1l2, 2 cos 6 = (2  k). (3.7.2)
Linearisation at (1,O) shows that this fixed point is a hyperbolic saddle point with
E;,,o, and E;;,, given by
u = u{  k  [k(4 + k)J1/*)/2 and o = u {  k + [k(4 + k)]'I2)/2, (3.7.3)
respectively, where (u, e) are local coordinates at (1,O). It is not difficult to then use a microcomputer to plot successive images of, say, one hundred points lying in a small interval of E:,, close to (1,O). The result of such a calculation can be quite spectacular (see Figure 3.33(a)). Of course, a suitable interval along E;,,,, can be iterated, using the inverse map
to complete the homoclinic tangle (see Figure 3.33(b)). Some uses of a computer program of this kind i re suggested in Exercises 3.7.3 and 3.7.4.
It is important to understand how the contortions of the stable and unstable manifolds influence the orbits of wandering points of (3.7.1). It is tempting to imagine that the latter also undergo wild oscillations but this is not the case. For example, the behaviour of (3.7.1) in the neighbourhood of the saddle point is
3.7 Homoclinic points 167
determined by Hartman's Theorem. Thus, since the eigenvalues of the linearisation at (1,O) are both positive (see Exercise 3.7.3), the orbits of individual points pass the saddle point as shown in Figure 2.l(e).
It is easily shown that the derivative map of (3.7.1) has positive determinant for all (x, y ) ~ W2 (see Exercise 3.7.5). A diffeomorphism, f: R2 + R2, with this property is said to be orientationpreserving (see Chillingworth, 1976, p. 139). A planar closed curve y can be oriented in two ways depending on whether an observer walking along the oriented curve finds the region enclosed by y on his right or lefthand side. When Det(Df(x)) > 0 for all x E It2, it can be proved (see Exercise 3.7.6) that the orientation of the image of y under f must be the same as that of y. Now consider the closed region So, with boundary yo, shown in Figure 3.34(a) and let yo be oriented according to the sense of description of the unstable manifold. It follows that the image of So under (3.7.1) must be one of the regions
Figure 3.33 (a) Approximation to the unstable manifold of a hyperbolic saddle point of the planar map (3.7.1) at (1,O) fork = 1.5. (6) Homoclinic tangle for (3.7.1) obtained by adding to (a ) an approximation to the stable manifold at (1,O). The latter is obtained by reverse iteration of a small interval of E;ls,, close to (1,O) (see Exercise 3.7.3).
168 3 Structural stability, hyperbolicity and homoclinic points
Figure 3.34 (a) Plot of the stable and unstable manifolds of the saddle point at (1,O) of (3.7.1). Since the map is orientationpreserving, the image of the manifold loop So must be one of loops Si , i = 1,2, . . ., with the same ori~tation as So. It is not dillicult to see that the orientation of the loops Si is opposite to that of So. In fact, for (3.7.1). the image of Si is Si+ , (see Figure 3.34(c)) but this is not the case in general. For example, the image of Si under the square of (3.7.1) is S, , , . (b) The images of So under iterates of the inverse of (3.7.1) are the regions S  i which wrap further around the fixed point at (0.0) as i increases. (c)
Numerical plot of the orbit of the point P = (0.64, 0.094) under (3.7.1). It sweeps around (0,O) twice, passing near to the saddle point on each occasion, before arriving in So at the fifteenth iteration. Subsequent iterates are carried away to infinity under the influence of the saddle point. Note that, since the manifold loops become extremely narrow and close together, the number of revolutions of the orbit about (0.0) before expulsion to infinity can depend sensitively on the choice or initial point.
3.7 Homoclinic points 169
Si, i = 1,2, . . . , with the same orientation as So, and not one of $, i = 0, 1,2, . . . , for which the orientation is reversed. Thus points in So are ultimately swept off to infinity under the influence of the saddle point at ( 1 , O ) . Similarly, points in So are swept around the fixed point at (0,O) and fed back into the vicinity of the saddle point once again. The role of this movement about (0,O) in the dynamics of (3.7.1) is best understood by considering images of So under powers of the inverse map. The preimages of So are a subset of the regions S,, i = 1,2, . . ., shown in Figure 3.34(6). Observe that, as i increases, these regions stretch further around (0,O). Indeed, for each NEZ' , there is an i (N) such that Sio, wraps around (0,O) N times. It follows that there are points in S,,, whose orbit makes N trips around (0,O) before it appears in So and subsequently sweeps out to infinity. It is not difficult to confirm these ideas numerically. An orbit exhibiting this behaviour is shown in Figure 3.34(c).
Similar orbits were shown in Figure 1.39 and 1.40 for the Htnon areapreserving map. This is no coincidence. Htnon has shown (see Hbnon, 1969) that every quadratic, areapreserving, planar map, with rotational linear part at the origin, is conjugate to the form (1.9.40). It is easily verified that the derivative of the map (3.7.1) has unit determinant for all (x, Y ) E R ~ (see Exercise 3.7.5)). Thus (3.7.1) and (1.9.40) must exhibit the same dynamics. For our present purpose, (3.7.1) has the advantage that the saddle point remains at (1,O) for all k , so that El;,,, and E;,,,, are easily calculated.
In the above discussion, we have assumed that the stable and unstable manifolds that intersect one another come from a single fixed point x*. Recall that Theorem 3.7.1 includes the case where the stable and unstable manifolds involved are associated with a fixed point of P. Similarly, if x* is a periodic point of period greater than one, then, for example, the unstable manifold of x* may intersect transversely with the stable manifold of f(x*) (see Figure 3.35(a)). Once again, the manifolds oscillate wildly because images of homoclinic points are homoclinic points. Given that the unstable manifold of f(x*) also intersects the stable manifold of x* transversely, then consideration of the images under f of a suitable parallelogram, R, again indicates that some power of f behaves like a horseshoe map (see Figure 3.35(b)).
This construction is also relevant to quadratic, areapreserving maps of the plane. Suppose x* has periodq and homoclinic points arise in the manner described above at each point of the periodic orbit, i.e. in the above argument x* t+ fim ''(x*) and f(x*)t+f'"'(x*), m'= m mod q, for m = I , . . ., q. Then we obtain a chain of homoclinic tangles as shown in Figure 3.36. In this case, the orbit of a point such as P in this figure could sweep around the whole periodic orbit before being fed back into the vicinity of x* at a different point, P'. Because of the massive stretching along the unstable manifold at each periodic point, the position of P' depends sensitively on that of P.
There is evidence of this kind of behaviour in the maps (1.9.40) and (3.7.1). The 'twodimensional' orbits shown in Figures 1.41 and 1.42 are associated with a hyperbolic periodic orbit, they are generated by iterating a single point and their
170 3 Structural stability, hyperbolicity and homoclinic points
extent is similar to that of the expected homoclinic tangles (see Gumowski & Mira, 1980, p. 303). In this situation, there is a good reason (see Figure 6.17) why orbits of this kind do not escape from the influence of the periodic orbit. Therefore, the plotted iterates of a single point appear to fill out the twodimensional region in an apparently random way.
3.8 The Melnikov function
In this section we describe a method for proving that transverse homoclinic points occur in the Poincark maps of certain types of flow in three dimensions. This
Figure 3.35 (a) Illustration of the unstable manifold of the periodic point x* intersecting the stable manifold of f(xt) transversely at XI and hence at infinitely many other homoclinic points. (b) The parallelogram R is iterated forward to R , and in reverse to R,. The map from F 2 to R , is horseshoelike.
3.8 The Melnikov finction 171
method is particularly interesting here because it can be applied to the Dufing equation which appears, numerically, to have a chaotic, attracting set (see Figure 3.26).
Consider the planar differential equation
x = fo(x) (3.8.1)
which has a hyperbolic saddle point at x = 0 and assume there is a homoclinic saddle connection, T, as shown in Figure 3.37. Now consider the product flow in W2 x S1 defined by
x = fo(x), 9 = 1. (3.8.2)
The saddle point of (3.8.1) at x =OeW2 becomes a periodic orbit y, = {(x, B ) E W2 x S1 I X = 0, OE S1) of saddle type. Moreover, the unstable manifold of yo, WU(yo), intersects the stable manifold, Ws(y0), in the cylindrical surface
Figure 3.36 Chain of homoclinic tangles that can arise on a hyperbolic periodic orbit.
Figure 3.37 Phase portrait for x = f,(x). The origin is a hyperbolic saddle point and r is a homoclinic saddle connection.
172 3 Structural stability, hyperbolicity and homoclinic points
x S1 E W 2 x S1 . This behaviour is nongeneric. In particular, the stable and unstable manifolds of the corresponding fixed point of the Poincarb map, Po, of (3.8.2) do not intersect transversely. The Melnikov method applies to small perturbations of (3.8.2) of the form
x = f , ( x ) + ~ f , ( ~ , ~ ) ; 8 = i (3.8.3)
with E E R+ and f l ( x , 0 ) = f,(x, 8 + 2n). For sufficiently small E, it follows from Proposition 3.2.2 that (3.8.3) also has a hyperbolic periodic orbit, ye, close to yo. However, the invariant manifolds, Wu(y,) and Ws(y,), need not intersect to form a cylinder (see Figure 3.38). The Melnikov function is related to the 'distance' between these two manifolds.
Let x 0 € R 2 be a point of the saddle connection I in the unperturbed system (3.8.1). Take a perpendicular section L to the saddle connection at x,. We use the point x, and the section L in the 8 = 0,plane, Zeo, as follows. Consider the perturbed system and the intersections of y,, Wu(y,) and Wyy,) with Zoo. This is equivalent to studying the Poincark map PLOD: Coo + Z% of the flow (3.8.3). will have a hyperbolic saddle point, x:,~, near to x = 0, with stable and unstable manifolds, WUs(x:eo) = WUs(y,)n C,o, which are close to r on Ceo (see Figure 3.39).
The distance between Wu(y,) and Ws(y,) on Zeo is calculated along L. Observe that this distance will, in general, change with 0,, since E > 0 implies that the curves W"(X:,~) and W"x:&) will be 8,dependent. Obviously, for the special case E = 0, the distance would be zero for all values of 0,.
Of course, the manifolds WU.'(x:d may intersect L many times, however, on
Figure3.38 Themanifolds Wu(y,)and Ws(ii)for (a)€ = 0 and ( b ) ~ > 0.
3.8 The Melnikov firnction 173
each curve there will be a unique point of intersection A"," closest to x, (see Figure 3.39). Let (xU.'(t; 0,, E ) , t ) , t E W, be the unique trajectory of (3.8.3) passing through A"" at t = B,, i.e. AuQ"s the point ~ ~ ~ ~ ( 6 ~ ; BO, E ) E Z ~ , . We then define the timedependent distance function,
A&$ 00) = fo(xo(t  00)) A CxU(t; 00, E )  xS(t; 00, & ) I , (3.8.4)
where x,(t) is the homoclinic trajectory of (3.8.1) with x,(O) = x,. In (3.8.4) the wedge product is defined by a A b = alb2  a2hl where a, b~ R2 have Cartesian coordinates (a,, a,) and (b,, b,), respectively. It follows that A,(t, 0,) is Ifo(xo(t  8,))l times the component of the vector [xu(t; 0,, E )  xS(t; 0,, E ) ] perpendicular to f,(x,(t  8,)). The latter vector is, of course, tangent to I at x,(t  8,). Thus, A#,, Oo)/lfo(xo)l is the distance between Wu(y,) and Wyy,) measured along L on
Coo. We can obtain a useful form for A,(@,, 6,) by studying (3.8.4) more closely. Let
x"(t; oo, E ) = x0(t  8,) + EX;(^, 8,) + 0 ( 1 : ~ ) (3.8.5)
and
where x;, x', are first variations with respect to E . Thus, (see Exercise 3.8.1)
t;"(t, 0,) = Dfo(xo(t  eo))~; .s( t , 0,) + f l(x0(t  Bo), t ) . (3.8.7)
Now define
A?(t, 0,) = f0(xo(t  8,)) A EX;"(^, 00), (3.8.8)
so that A,(t, 0,) in (3.8.4) can be written in the form
Figure 3.39 The intersections of y,, Wu(y,) and W'(y,) with 2,, for E = 0 and E z 0.
174 3 Structural stability, hyperbolicity and homoclinic points
We can obtain differential equations for A: and A:. It can be shown that, since io ( t  0,) = fo(xo(t  8,)),
&(t, 0,) = ~CTr(Df~(x,(t  90)))fo(xo(t  9,)) A x;(t, 0,)
+ fo(xo(t  8,)) A f,(xO(t  o,), t ) ~ . (3.8.10)
The expression (3.8.10) is greatly simplified if f, is a Hamiltonian vector field, as it is for the Dufing equation, for then Tr(Df,(x)) = 0 (see (1.9.21 and 24)) and
Axt , 9,) = ~f,(x,(t  80)) A f,(x,(t  eO), t) . (3.8.11)
Integration of (3.8.1 1) from t =  co to t = 8, gives
00
A:(O,, 9,) = E 1 &,(x,(t  0,)) A f,(xo(t  8,), t ) dt . (3.8.12)  m
Here we have noted that A:( co, 8,) = 0 because x,( m) = 0 = fO(0). A similar calculation leads to
At(e,, 9,) =  E f0(xO(t  0,)) A f,(x,(t  0,), t ) d t . (3.8.13)
and therefore,
W m
A.(B,, 0,) = E 1 fo(xo(t  8,)) A f,(x,(t  O,), r ) dr + 0(c2). (3.8.14)  m
Finally, we define the Melnikoo function, M(Oo), by
so that
A,(8,, 6,) = EM(O,) + O ( E ~ ) . (3.8.16)
Proposition 3.8.1 If M(8,) has simple zeroes, then, for sufficientb small E > 0, Wu(x$) and WS(x;) intersect tramwrsely for some 0, E [0 ,2n) On the other hand, if M(9,) is bounded away fiom zero, then W " ( X ~ ~ , ) ~ W'(X:,~) = 125 for all 8,.
In allowing 8, to vary, we are effectively taking a fixed reference point x, and section L, perpendicular to f,(x,), in each section Ceo, 8 ,~[0 ,2n) . By taking E sufficiently small, A,(@,, 8,) is an arbitrarily small perturbation of EM(@,). It follows that if eM(8,) has a simple zero then so does A#,, 0,). This means that there is a value, 8, of 8, at which A,(9,, 0,) changes sign, corresponding to xu(O0; O,, E ) xa(9,; go, E ) reversing its orientation relative to fo(xo). Clearly xu(@; @, E ) =x1(@; @, E ) and, therefore, the manifolds Wu(xZ,) and Ws(x2,) of the fixed point x:, of the Poincark map PC,, intersect transversely on L near to
3.8 The Melnikov function 175
x,. Of course, all the Poincark maps P,,%, O,E [0,21r), are topologically conjugate (see Exercise (1.7.3)) and, consequently, WU(xtoo) and W"X:,") must intersect transversely for all O o ~ [ O , 271) (although, obviously, not always near to x, (see Figure 3.40). Equally, if M(0,) is bounded away from zero, then, for sufficiently
Figure 3.40 The manifolds WU(y,) and WS(y,) intersect in a homoclinic trajectory that ultimately approaches y, as t + co. When A,(Oo, 0,) has simple zeroes, this trajectory passes through the section L x LO, 2n) at least twice. An impression of the nature of the homoclinic trajectory can be gained by recalling that 0 = 2n is to be identified with B = 0. Thus the segment BOB1 continues as B,B, and CoCl as CIC,. The trajectory itself is (Be ,B.)u(C, ,C,). It follows that corresponding pairs of zeroes occur for any choice of x, E r. Moreover, if M(Bo) is bounded away from zero on [0,2n), then it is so, independently of the choice of x,, and no homoclinic points occur. For given O,, the stable and unstable manifolds of the fixed point xz, of the Poincark map P,, are obtained by taking the corresponding section in this figure.
176 3 Structural stability, hyperbolicity and homoclinic points
small E, SO is AE(O,, 0,). This, in turn, means that transverse homoclinic points do not occur on L for any 0, E [O,2n). As Figure 3.40 shows, this conclusion does not depend on the choice of x , e r through which L passes. Hence, there are no homoclinic points.
Example 3.8.1 Show that the Poincari map of the Dufing equation
1 = y, j=xx3+&(acoseby) , = 1, (3.8.17)
a, b > 0, has transverse homoclinic points, for sufficiently small values of E, provided
Solution. When E = 0, (3.8.17) becomes
i = y, j = x  x 3 , O=I, (3.8.19)
so that fo(x) = (y, x  x ~ ) ~ . The differential equation ir = fo(x) has a hyperbolic saddle point at x = 0 and two further fixed points at x = (+ 1, o ) ~ . It is a Hamiltonian system with
and the level set of H ( x , y) = 0 consists of two homoclinic orbits, T$ ,and the saddle point at x = 0 (see Figure 3.41). It can be shown (see Exercise 3.8.2) that the trajectories passing through (x, y) = (+ (2'12), 0) at t = 0 are given by
(~ ' ( t ) , yf ( t ) ) = (b(2Ii2) sech t, T(2LiZ)~ech t tanh t). (3.8.21)
Figure 3.41 The phase portrait for the planar system x = &,(x), f,(x) = (y, x  x ~ ) ~ . Stable and unstable manifolds of saddle point x = 0 coincide to form a pair of homoclinic orbits r$. The level set H ( x , y)=O is r;~{ojur,+.
3.8 The Melnikov function 177
Comparison of (3.8.17) with (3.8.3) gives
which satisfies fl(x, 0) = fl(x, 0 + 2n). It follows (from (3.8.15)) that the Melnikov function for the homoclinic orbit T,+ is
+ 21i2b sech(t  0,) tanh(t  e,)] dt.
The change of variable of integration t c* t  0, gives
a0
 2b sech2(r) tanh2(t) dr. (3.8.24)
The latter integral is easily evaluated, while the former can be simplified by writing cos(t + 0,) = cos(t) cos(0,)  sin(t) sin(0,) and noting that
because the integrand is an odd function oft. Thus,
The integral occurring in (3.8.26) can be evaluated using the method of residues (see Exercise 3.8.4)) and we finally obtain
Clearly, if (3.8.18) is satisfied M(0,) has simple zeroes and, by Proposition 3.8.1, transverse homoclinic points must occur. On the other hand, if the reverse inequality is satisfied, M(8,) is bounded away from zero and Proposition 3.8.1 implies that there are no homoclinic points.
There is one remaining possibility for the system (3.8.17): namely that
In this case, M(0,) has a double zero at 8, = 3x12. This corresponds to W " ( X ~ ~ ~ ~ ) and W'(X:,,~,) meeting tangentially rather than transversely. As before, the orbit of such a homoclinic point under P,.,,,, consists entirely of tangential intersections
178 3 Structural stability, hyperbolicity and homoclinic points
Figure 3.42 (After Ueda, in Guckenheimer & Holmes, 1983, p. 192.) Stable and unstable manifolds for the Poincak map of the Duffing equation (3.8.17) with ~b = 0.25 and (a) m =0.11; (b) &a = 0.19; (c ) &a = 0.30. Observe the tangency of the stable and unstable manifolds in (b).
180 3 Structural stability, hyperbolicity and homoclinic points
of W"(X~,,,~) and WS(x:,,,,). Moreover, since P,,,, and PcTe0 are topologically conjugate for all 8, and elo, these homoclinic tangencies occur in all P,,,,. Ueda (see Guckenheimer & Holmes, 1983, p. 192) has computed stable and unstable manifolds for the hyperbolic saddle point of the Poincari map of (3.8.17) and some of his results are reproduced in Figure 3.42. It is not difficult to verify that the value of a/b at which homoclinic tangencies occur numerically (see Figure 3.42(b)), is in close agreement with (3.8.28).
The occurrence of homoclinic tangencies has important repercussions the details of which are beyond the scope of this text. Newhouse (1979, 1980) has shown that if such a tangency occurs at x, for feDiffr(W2), then there is an 1 EC'close to f for which tangencies also occur stably in a hyperbolic invariant set. This set lies near to the orbit of x, and is known as a wild hyperbolic set.? also has an infinite number of stable periodic orbits  or 'infinitely many sinks'  as the title of Newhouse's original paper had it. We refer the reader to Guckenheimer & Holmes, 1983, pp. 33140 for a more detailed description of these ideas. However, this kind of behaviour may occur in Ps,, for a/b near to the critical value (3.8.28).
As we have already noted (see Figure 3.26), numerical approximations to (3.8.17) exhibit a complicated attracting set. Such a set appears even in the Euler approximation and it is then not difficult to verify that ajb must exceed a critical value before it appears. This suggests a connection between the attracting set and the occurrence of homoclinic points. Indeed, the careful numerical work of Ueda (in Guckenheimer & Holmes, 1983, p. 90) (see Figure 3.43) has led to the conjecture that the attractor is the closure of the unstable manifold of the saddle point. While this can be justified for ajb less than the value in (3.8.28) (Guckenheimer & Holmes, 1983, p. 91), the situation is more complicated when homoclinic points are present.
Exercises
3.1 Structural stability of linear systems 3.1.1 Considera real, n x n matrix, A, with eigenvalues A,, . . ., L, that are not necessarily
distinct. Let B, with eigenvalues p,, . . ., p,,, be cclose to A in L(RR). The spectral variation of B with respect to A is defined by
S,(B) = max [min (Idi  PA)]. (E3.1) i i
(a) Assume that A can be diagonalised and show that
S,(B) < c.
(b) Suppose A cannot be diagonalised and show that
SA(B) < (w)""
provided EK l/n. (c) Deduce that {p,, . . . . A) + {Il, . . ., A,} as c r 0 for any A E L(W").
3.1.2 Let SD(RN) be the subset of structurally stable linear di&omorphisms in L(RR). Show that a linear diNeomorphism is structurally stable if and only if it is hyperbolic. Hence, or otherwise, show that SD(Rm) is open and dense in L(R").
Exercises 181
3.1.3 Let S be the subspace of L(Mn) defined by {(A y)li + 0 or I}. Show that every .\ r ,
linear difieomorphism in S is structurally stable within S but not within L(R2).
3.1.4 Consider the subspace, 0(M2), of L(RZ) defined by (AIATA = 1, A E L(W2)}. Show that no element of 0(R2) is structurally stable in L(R2).
3.2 Local structural stability 3.2.1 Let the vector field x(x)~Vec'(U), Ug R" and open, have a hyperbolic fixed
point at x* = O E U and suppose that X(x) is_ an Ec'perturbation of X. Verify Proposition 3.2.1 for the special case when X  X is (a) constant; (b) linear; (c) O(lxlk), k 2 2.
3.2.2 Find q =q(c) such that each of the following vector fields is cC'close, on U = ((8, r)lr < 21, to.# = r(1 r), b = 1 ; (a) i = r(1 + q  r), O 7 I ; (b) i = r(l  r + qr2), 0 = 1 ; (c) i = (1 + q)r(l  r), O = I. Verify that the flows (a j ( c ) all have a hyperbolic periodic orbit near to r = 1 for sulliciently small values of E .
3.2.3 Show that the nontrivial fixed point x* = (c/l, a/b)' of the VolterraLotka vector field
a, b, c, f > 0, is nonhyperbolic. Find a first integral for the system x = X(x) and determine the topological type of x*.
Consider vector fields of the form X + X, on a disc of radius R > (x*l, where: (a) X,= (6x, 6y)'; (b) Xa= (ax2, O)T. Choose 6 in each case such that IIXalll < E . If c is sulliciently small, show that X + X, has a fixed point y* near to x* for both perturbations but that the topological type of y* is the same as x* for (a), while it is different for (b). Explain why this result is consistent with Proposition 3.2.1?
3.3 Flows on twodimensional manifolds 3.3.1 All of the following vector fields are structurally unstable on R2. TO which of
these examples does Theorem 3.3.1 apply? Use the theorem, where applicable, to explain the nature of the instability. For the remaining examples construct ~C'close systems to exhibit their structural instability in Vec1(9), where 9 is the closed disc of radius 2 centred on the origin. (a) i = r(r O = 1; (b) i. = r(l  r), 0 = sin2(U); (c) i = 2y(l x2) + xB(x), j = 2x(1 y2) + yB(y); where B(x)=exp{x2/(1 x2)) for 1x1 < 1 and =O for 1x1 3 1.
3.3.2 The flows 9,: R2 4 R2 of the following systems give rise to flows on the torus T2 = {(O,, 02)10 G O,, O2 < I ) by taking mod 1 in both components of 9,. Use Peixoto's Theorem to show that these toral flows are structurally unstable. (a) 1 = sin(2nx). j, = 0; (b) i= 1, j=2 . Illustrate these instabilities by giving topologically distinct systems which are &<'close for arbitrarily small E.
3 Structural stability, hyperbolicity and homoclinic points
Consider the system on the cylinder C = ((0, r)10 < 8 < 2n, rE R}, given by
Show that it is structurally stable on every set S, = ((0, r)l n < r < n}, n~bZ+, but that it is not structurally stable on C.
3.4 Anosov diffeomorphim
Use Peixoto's Theorem to show that none of the following diNeomorphisms / E DiN1(S1) are structurally stable: (a) /(O)=(O+a)modl,acO; (b) f(O)=(O+a)modl,aeR\Q; (c) f(0) = (0 + sin2(2nO)) modl ; (d) f(0) = (0 + + 0.1 sin2(2nO)) modl.
Show that all of the periodic points of an Anosov automorphism f: Tn + TT" are hyperbolic. Let B* be a periodic point off of period q € Z + . Give expressions for the stable and unstable manifolds of the periodic orbit containing B*.
Let the Anosov automorphism f: T2 + T2 have lift A: R2 + R? given by (3.4.12). Prove that f has periodic points of every prime periodq by showing that there exists x€R2 such that Aqx  xeZn, where x does not represent a fixed point.
Let f and g be Anosov automorphisms of T" with lifts given by automorphisms A, B: R" + R". Prove that A and B are similar matrices iff and g are differentiably conjugate. Conversely, show that, if A, B are lifts of Anosov automorphisms f, g and they are similar by a matrix C with integer entries and determinant + I , then f and g are dikrentiably conjugate.
Show that the Anosov automorphism f induced by
has a saddle point at n(O), where n: R2 + T2 is the map given in (3.4.2). Find the equation of the separatrices of A at 0 and show that they have irrational slope. What are the implications of this on the torus? Show that the point on the torus given by n(xt), where xt = A n o s o ~ a u t o ( l 3 ~ ~ ~  1)/2(13~/~), 1/13112), is a transverse homoclinic point. How does the Anosov automorphism considered in this question diNer from that given by (3.4.12)?
3.5 Horseshoe diffeomorphisms
35.1 The canonical example
Obtain explicit equations for the horseshoe map r f: Q + R2 on its restriction to
Pow PI c Q (see Figure 3.13). Verify that (x, y ) ~ (f1(x),f2(x, y)).
Let A be the Cantor set of the horseshoe diNeomorphism f as defined in (3.5.4). Complete the proof of Proposition 3.5.1 by showing that A is invariant under fI. Hence show that f(A) = A.
Exercises 183
3.5.3 Let g: SZ r S2 be the globally extended horseshoe dilfeomorphism. Prove that:
(a) Q(') is invariant under g and n Qw) is invariant under gI; ns N neZ '
(b) n Q(#) ( 0 Q("') is the inset (outset) of A for g on Q. IIEN nsZ
3.5.4 Suppose g: Q + R2 is the horseshoe map and take xo, with coordinates (x,, yo), to be a point of Q\A. Let d = d(xo, A) = min {lxo  XI} be the horizontal distance
(X .Y )€A from xo to A. Find the maximum value, N ( d ) , of n such that f"(xO)cQ.
35.2 Dynamics w symW sequences 3.5.5 Let Z, be the set of all biinfinite sequences on the m symbols S = {0,1, . . ., m  1)
and a: I;, + Z, be the left shift a(a), = an, (see (3.5.7)). Prove that: (a) a has periodic orbits of all periods as well as aperiodic orbits; (b) there is a natural topology on & for which the periodic points of a a n dense
in Z,; (c) there exist dense orbits of a on Xs.
3.5.6 Let a (B) be the left (right) shift on the symbol sequence space X. Show that: (a) a is a homeomorphism and a ' = B; (b) a is topologically conjugate to P.
3.5.7 Let a: Xs + Z, be either a left or right shift on &, the space of symbol sequences with S = {0,1,. . ., m  I}. Show that there are mqsequences a such that cr'(a) a . Let m,be thenumberofperiodkpointsofaand K = {kJkeZ' &kJq}. Prove that
x mk=mq (E3.5) ksK
for m, q€Z+ . For m = 2, use this result to find the number of period12 orbits.
35.3 Symbolic dynrmirs for the b m h o e diffeomorpb'm 3.5.8 Let f: Q . R2 be the horseshoe map. Use (3.5.3) to prove that each vertical strip
of Q'"I, ne N, can be described by a binary sequence of nsymbols.
3.5.9 Let (x, y)€A, the invariant Cantor set of the horseshoe map I: Q+ R2. Define {a,}, {bi}i", , as follows.
(i) Consider the nested vertical strips given by n Q(". 1fXc {;: then a, = {; :. n t N
of the previous strip in the nesting. right
(ii) For the nested horizontal strips n QU), define b1 = {': if {:: and
bi = { z :. i 5 2, if y lies to the of the previous strip in the nesting.
Show that each point (x, y ) of A can be written in the form
Hence show that the subset of A in the quadrant x, y > 0 is homeomorphic, by the fivefold magnification (x, y ) ~ (5x  2,5y  2), to the whole of A. What does this imply about the structure of the set A?
184 3 Structural stability, hyperbolicity and homoclinic points
3.5.10 Use the form for X E A given in Exercise 3.5.9 to find the coordinates in Q of the fixed and period2 points of the horseshoe map f: Q + R2.
3.5.11 Let f: Q+ W2 be the horseshoe map. In the notation of 53.5.3, the set N
A(N' = n Q@) consists of21N connected components, each one being a square m =  (N I )
region of side 215" lying within Q. (a) Prove that (3.5.17) uniquely associatesasymbol block dN'= ,,, . . . , aN)
with each connected component, ~ ( 0 ~ ~ ) ) say, of AtN'. Locate ~(cr '~ ') for the following symbol blocks dN':
i ) 1 . I}; (ii) 1 I } (iii) (010 101).
(b) Let q(N) = {q (N I ) r . . ., qN) and dN' = { v  ( ~  ,), . . ., vN} be two symbol blocks of length 2N. Explain how to choose a point XEA in ~ ( q ' ~ ' ) such that fZN(x)e K(v(~)). Hence show that there exist points in ~ ( q ' ~ ' ) whose orbit under fZN visits every connected component of AtN) in any desired order.
(c) What restriction must be imposed on the elements of the sequence a if the orbit of the point x = L(u)E A is to remain in a particular component, K(u'~') say, for k applications of f? What is the maximum number of connected components that can be reached from ~ ( a ' ~ ' ) in k iterations off?
(d) What aspect of the dynamics of flA do the observations (a)(c) reflect?
3.5.12 Recall that the horseshoe map f: Q + W2 satisfies (x, y)C* (j,(x), fz(x, y) ) . Verify this property for flA by considering the left shift a: C t I: that is conjugate to f. Show that j,: [I, 11 + R has a repelling invariant Cantor set.
3.5.13 The Baker's transformation B: T2 t T2 is defined by
for (xmodl, y m o d l ) ~ T, (Arnold and Avez, 1968, Appendix 7). Describe the effect of this transformation on the rectangles Po = lo,$) x [0, I)
and PI = [f, 1) x [0, 1). Show that every point X E TZcanbewritten in the form
whereo = {om},"= , is a biinfinitesequence of{0,1). Use this result to show that
where a: C + C is the left shift. Prove that h: C. T2 is not onetoone by finding h(a,) and h(o,), where
a , = { b l .O ') and a, = ( '10.0 '1 ( 'i and i ' indicate indefinite recurrence of the symbol i to left and right, respectively). What is the general form of points X E T2 for which h fails to be injective? Given that these problem points can be disregarded (see Arnold & Avez, 1968, p. 125), show that the nonwandering set of B is the whole of T,.
3.6 Hyperbolic structure and basic sets
3.6.1 Let g: R2 + Rz, given by g(x) = (g,(x), g2(~))T, x = (xl, x , )~ , be a smooth map and y : (  1 , l ) + RZ be a smooth curve. Show that the tangents to the curves y
Exercises
and gay at t = 0 are related by the equation
where Dg(x) is the matrix of partial deivatives (%)) . Illustrate this result \ax,/,,= I
by finding the image under g(x,, x,) = (exp(xl + x,), X,X,)~ of the curves: (a) x, = t, x2 = t cos t ; (b) x, = t + t2, xZ = tan t.
Show that the curves (a) and (b) have a common tangent at x = 0. Verify that the image curves under g also have a common tangent that is given by (E3.7).
3.6.2 Consider the representation of S1 shown in Figure E3.1. Find the overlap map h,, between U, and U,.
The norms II.II,, on TU,,, and 11. I Ix2 on TUzXl are compatible if
Ilvllx, = I I ~ ~ I ~ ( X ~ ) V ~ . ~ . (E3.8)
(a) Prove that the Euclidean norms on U, and U, are not compatible. (b) Find the norm on U,\P, which is compatible by h, , to the Euclidean norm
on U,. Show that this norm cannot be extended continuously to the whole of u,.
(c) ve& that, at each point xi of Ui
is a positive definite inner product on TU,,. Show that the nonns /lulli = ( v , v ) ~ are compatible.
3.6.3 Let M be a differentiable manifold and x* be a fixed point of the diffeomorphism f: M , M. Let h: W c M + U be a chart at x* and h(x*)= ji*. Show that the eigenvalues of the derivative D(hfhI)(%*) are independent of the choice of the chart (U, h) at x*. What is the significance ofthis result in relation to the problem of defining a hyperbolic fixed point on M?
3.6.4 Show that the Anosov automorphism f: T2 + T2 given by A: RZ P R2, where
A = (: i), satisfies the hyperbolicity conditions (36.9 and 3.6.10) at each point
of R = T2.
Figure E3.1 Stereographic projection from (S1\PZ) 4 U, and (S1\P1) + Uz provides an atlas for the unit circle S' c R,.
186 3 Structural stability, hyperbolicity and homoclinic points
3.6.5 Consider the diffeomorphisms g , , g,: S2 + St defined in Figure E3.2. Outline arguments to show that g,, i = l , 2 , has an invariant Cantor set A, r Q such that gJAi is conjugate to a shift on m(i)symbols, where m(1) = 3 and m(2) = 4. What are the basicsets ofg, and g,?For both maps,draw schematicdiagrams illustrating the dynamics of the wandering points. Is g,lA, conjugate to g,(A,?
3.6.6 Find which of the following diffeomorphisms of the torus T Z satisfy the hypotheses of Theorem 3.6.1 and describe their basic sets:
(a) f (n(x)) = %(Ax), A = (: 3; Figure E3.2 The restrictions of g, and g, to the capped square Q' are shown in (a) and (b), respectively. On each component of QIO', i = 1, 2, &. is assumed to be linear. The map gl is a contracting diffeomorphism on both F , and G, with hyperbolic fixed points P, EF; =g,(F,) and P,EG; =gl(G,). Observe that G;=g,(G,)c F, and g21F2 is a contracting diffwmorphism with hyperbolic fixed point P, E F; = g,(F,). On S2\Q', both g, and g2 have a single repelling hyperbolic fixed point Po. It may be assumed that Theorem 3.6.1 applies to both g, and g,.
Exercises
(c) f,(n(x)) = ~ ( Q ~ ( x ) ) , where Q , is the timeone map ofthe system (i) = (::::;~). 3.6.7 Obtain the Hknon attractor usinga microcomputer to plot the iterates ofthe map
x , = y  1 . 4 x 2 + l , y1=O.3x, (E3.9)
with initial value (1,O). Choose x and y scales such that the square Q = {(x, y ) ( l ~ ~ < 1, < I } fills a large portion of the screen. Observe the braided nature of the attractor by magnification and shifts of the origin.
3.7 Homoclinic points 3.7.1 Use the explicit form of the horseshoe map I: Q + R2 on P O u P l (see Exercise
3.5.1) to locate the fixed points off and their stable and unstable manifolds. Show that there exist transverse homoclinic points at (x, y) = (4, 4) and (4,f). Hence obtain Figure 3.31. Show that any homoclinic point xt of a periodic orbit on A is itself an element of A.
3.7.2 Let f: M + M satisfy the requirements of Theorem 3.7.1, i.e. f is a KupkaSmale diffwmorphism with a transverse homoclinic point associated with one of its
P  1 periodic points. Show that ;i = U e(A) is a Cantor set such that f ( i ) = A.
I=0
3.7.3 Find the eigenvalues of the linearisation of (3.7.1) at the fixed point (x, y) = (1,O) and verify that both are positive fork > 0. Show that the eigendirections are given by y = { k f [k(k + 4)I1l2)(x  1112. For k = 1.5 take an interval of approximate length 0.0001 containing 100 points on the appropriate branch of the unstable manifold of the saddle fixed point. Plot 15 iterates of each point under (3.7.1) to obtain a numerical approximation to the unstable manifold at (1,O). Use the inverse map (3.7.4) to complete the homoclinic tangle shown in Figure 3.33. Modify the program toexhibit the image oieach successive iteration ofthe interval separately. Observe the repeated stretching and folding around the origin.
3.7.4 Use the program developed in Exercise 3.7.3 to study how the extent of the homoclinic tangle depends on k . Plot the tangle for k = 0.4,0.8, 1.2, 1.6 and 2.0. Comment on your results.
3.7.5 Show that the derivative map of (3.7.1) has unit determinant for all (x, y)eR2. Given that there exists a linear conjugacy between (3.7.1) and (1.9.40), find a relation between k and a. Modify the program used in Exercise 1.9.9 to generate orbit plots for (3.7.1) corresponding to Figures 1.38 and 1.39.
3.7.6 Let y(t), t ~ l c R, define a closed curve, y, in the plane oriented with increasing t. Suppose that T(s), s~ J c R, defines a segment ofa planar curve, T, that intersects y transversely. Assume that the point of intersection is given by x, = y(0) = T(0) and that T(s) lies inside y for s > 0. Verify that y(0) A T(0) determines the orientation of y.
Let f: R2 + RZ be a diffeomorphism and show that
(~:Y)(o) A (f:r)(o) = Det D~(x,)[~(o) A r(o)] (E3.10)
gives the orientation of f.y.
188 3 Structural stability, hyperbolicity and homoclinic points
3.7.7 Let I: Q + R2 be the horseshoe map and x* be a periodic point of period q > I, on the invariant Cantor set A. Use symbolic dynamics to construct a point xt of A which is homoclinic to the periodic orbit containing x* (cf. Figure 3.35). Can the transverse nature of the homoclinic point be detected by the symbolic dynamics?
3.8 The Melnikov function 3.8.1 Consider the solutions of
given by (3.8.5) and (3.8.6). Obtain (3.8.7) by substitution and comparison of order E terms. Hence deduce (3.8.10).
3.8.2 Show that H(x,  x2 + $x4) is a Hamiltonian for the system k = y, y = x  x' and verify that the level set H = 0 consists of a saddle point at x = 0 and two homoclinic orbits TG and Ti given by
(~'(t), y* (t)) = (+2'" sech(t), ~ 2 " ~ sech(t) tanh(t)).
3.8.3 If the system
x = fo(x) + & f , ( ~ , t),
where f, has period 2n/w, has a homoclinic orbit xo(t), for E=O, then the corresponding Melnikov function is
Oo~[O, 2x1~). Use To+ obtained in Exercise 3.8.2 to find a Melnikov function for the system
+ m
38.4 Prove that [ mh(t) fanh(t) in(wt) dt = nw sh (no /2 ) by using contour inte J  m
gration on a rectangle in C with vertices at ( + R , 0), (+ R, in)andletting R * oo.
3.8.5 The Melnikov function given in (3.8.1 5)can also be used to indicate the separation of stable and unstable manifolds of two different saddle points in a Hamiltonian system (the socalled heteroclinic case). Show that the saddle connections, rg ,. between the fixed points (n, 0) and (n, 0) of the system
= y, $ = sin(x) + &(a  by) (E3.15)
with E = 0, are given by
Calculate the Melnikov function for (E3.15) along these orbits and show that
for To+, T,, respectively. Explain why M(0,) is constant.
Exercises
3.8.6 The SineGordon equation
k=y, j = sin(x) + &(a cos(wt)  by), (E3.17)
a, b > 0, with E = 0, has saddle connection orbits, I$, between lixed points at (f R , 0). Calculate the Melnikov function for (E3.17) along Tf and show that it can be written in the form
Describe the regions of the (a, b)plane for which transverse hetcroclinic points occur.
Hints for exercises 395
1.4.6 Polar coordinates in the x, f plane give
HINTS FOR EXERCISES
Chapter 1
W, = {exp(ix))a < x < b), W2 = {exp(ix)lc < x < d) such that W, u W2 = S', (b  a), (d  C) < 2n. Cmoverlap maps.
hb.f.hil = (hphi ')(hd.f.h;')(h,.hil). Composition of two Ckmaps is and overlap maps h,,h;l and h,.h; ' are Ck since r >, k . Differentiability of f is independent of charts.
(a) Pick open subsets A, B, C, D of R2 such that {n(A), x(B), n(C), n(D)} is an open covering of T 2 and restrictions of x to A, B, C, D are homeomorphisms.
(b) Wl = S2\N, N the north pole; W2 = S2\S, S the south pole, h,(h,) is the stereographic projection from N ( S ) poles. Overlap map (r, c p ) ~ (4/r, cp), r # 0.
Arnold, 1973, pp. 1635.
(a) yes; (b) no; (c) circle map not homeomorphism.
Fixed points x = 0.4; all other points period2.
Plot y = f 2(x).
f a n orientationreversing homeomorphism on W implies it is strictly decreasing. fix + I) =f(x) 1 _as in proof of Proposition 1.2.1. Fixed points of j only at intersection of y = j ( x ) with y = x and y = x + 1 (see Proposition 1.2.2).
Definition 1.3.1 implies Q, is C1 for all t~ W. (p;' =(p,.
(a) i = x3; (b) i = x, j; = yZ
Minimal: (a) S f ; (b) {x, R,,,(x), . . ., R:l,'(x)}, XES ' . General: (a) S'; (b) S = UUR,,~(U)U.. . R;l,'(U), U E S1, closed.
(a) Show Rc is open.
Separatrices connecting n = 1, 2, 3, 4 saddle points enclosing unstable focus. Consider Hamiltonian system with desired saddle connection and introduce dissipation in the region bounded by the separatrices, e.g. i =  2y(l  x2) + /~vB(x), j = 2.41  y2) + pyB(y), p > 0,
Construct positively invariant set containing no lixed points.
1.5.1 (i) Similar construction to Example 1.5.1 for topological conjugacy with reference intervals [1,2] for / and [1,8] for g.
Conjugacy preserves fixed points.
Use Proposition 1.2.2. Plot = p ( x ) . Conjugacy preserves periodic points.
Use (1.5.1 1).
Differentiate (1.5.1 1) and set x = 0.
If x* periodic with periodq, p ( j ) = ~ i m ( T ( x * )  x*)/nq mod 1. (... ) Consider separatrix of the saddle which is of oppositestability to that of the node.
Consider the lifted flow Q,(x, y) = (x + t, y + at) on W2. Periodic orbits are given by Qr(x, y) = (X + m, y + n), T # 0, m, ~ E Z .
Recall that (Q x +),(x, y) = ((p,(x_), +,(y)l, X E M, y E N. Consider the lifted flows @,(x) = x + 1, &(x) = X + 21'2t, *,(x) = *;(x) = @,(x).
Arrowsmith & Place, 1982, $2.3.
rpl(x)=xe/[xe.T t I].
Fixed point x* must lie in X but X(x*) =0. Show that P2((pto(x))= (p,(Pl(x)), xES,.
Cylinder. Two limit cycles: stable x =O; unstable x = 1.
(a) Mobius band; (b) Klein bottle. x(t) = C exp(t  cos t).
IfQ(t) is a fundamental matrix so is Q(t + T) and Q(t) = Q(t + T n  '(to + T)Q(to). All Po0 are conjugate.
1.8.5 (a) ~ ( t , to) = l 7 exp:i,J, 7 = t to;
1.8.7 Use polar coordinates. Null solution is stable
s ~ n t cos t sin t
1.9.1 Sketch level curves of H(x,, x,).
1.9.3 The generic case has nonzero eigenvalues.
1.9.4 Hamilton's equationsin plane polarsarei. = r  ' dH/2O, d =  r  ' dH/dr. Examine extrema of H as a function of r for various fixed values of 0.
396 Hints for exercises
ZH aH an ay 1.9.7 I f i =  a n d j =   then = ,
3~ ax a . ~ ay 1.9.8 Y is symplectic to order 111.
Chapter 2 2.1.1 If the Jordan form of L is not diagonal examine the powers of blocks of the form
i.1 + N, Nij = Ji,, ,. Observe ( N ' ) ~ ~ = a,.,,, I 6 k 6 n  1, Nu = 0.
2.1.2 (a) p = max{ll.,l,. . ., Ii,I}, (b) Pick N > 3 such that p = NrlN(il < I.
2.1.3 (i) A l p : ue(l) 5"2 u, orientationpreserving expansion;
A l e : .++(!?)a, orientationpreserving contraction;
(ii) AIEu: ut+ (1 + 2'12)u, orientationpreserving expansion; A1 E': s (I  2'12)o, orientationreversing contraction.
f  f 0
2 . 4 Real Jordan form of A is (  :I. A l p is a rotational contraction.
2.1.6 x = Ax is linearly conjugate to $= Ay, A = [l,6,j]:j=,. Show that ji = i iyi is topologically conjugate to ii = sign(ii)zi, i = 1, 2, 3, and use Exercise 1.6.5.
2.1.7 Use Theorem 2.1.2.
2.1.8 dim F + dim En = n and restrictions to F and Eu may be orientationpreserving or reversing.
2.2.1 (a) Df(O,O)= (i i ) , saddletype with reflection;
(b) Df(O,O)= [ :). expansion with reflection.
2.2.2 DQ,(O) = exp(DX(0)) = exp 3 has eigenvalues msh(l) sinh(1). Show that
W;."(O) = W";"O), where Q is the flow of k = y, y = x x2. Obtain W",O)n W:(O) from a first integral.
2.2.3 If P(x) = y then fql U and PI V are conjugate by P .
2.2.4 If y E Ws(f'(x*)) then Lim V k ( y ) = P+'(x*), k =0, . . ., q  I. No, construct ' .a
counterexample: (a) lor periodI, consider Q, in Figure 1.16, let f = Q, and observe y E A$ Wp(P0)
but Po c L,,(y); (b) for periodq > I, construct periodic orbit in similar manner to Exercise 2.2.5.
Hints for exercises 397
2.2.5 Vector field is symmetric under clockwise rotation by 4 2 . A fixed point x* of (p,,
with topological type given by DQ,(x*) = exp(DX(x*)), becomes a periodic point off =Q, .R , , .
2.2.6 Let x, = cp,,(x,) and define Sb = Q_,(S,). Use flow box coordinates to prove Po: So +So and Po: So + So are C'conjugate and result then follows from Exercise 1.7.3.
2.2.7 Introduce cylindrical polar coordinates and recognise closed orbit for r = (x: + xi)'1' = 1, z = x, = 0. The Poincart map cp,, defined on the plane 8 =constant has a fixed point at (r, z) = (I, 0). Hyperbolicity follows from DQ,,(~, 0) = exp(DX(1,O)) and Hartman's Theorem.
2.3.1 (i) Solve quadratic for y and expand square root. (ii) Use (i) to obtain y, and substitute into expansion for (I + y,) '
2.3.4 Let h,B)=( ~ I Y : + a z Y ~ ~ z + a 3 ~ : , write down LAh,(y) and show that ai and
~ I Y : + ~ Z Y ~ Y , + b 3 ~ : bi, i = 1,2,3, can be chosen such that LAh,(y) = X,(y). Find a, = a , = a, = b, =
2.3.5 Use resonance condition to show that (2) is the only resonant term.
2.3.6 Use resonance condition and q l , + p i , = 0.
2.3.7 Use resonance condition. Normal form when i, = mi,, m > 2, f x;~,+;$)
2.3.8 Matrix representing LA is triangular with repeated eigenvalue 1. Since A, = A2 = A, Am.i = A for all m, i.
x7x;* 2.3.9 Use basis {(:), . . ., ( xl;xym O ), . . ., (:),(:), . . ., ( ), . . .. (:)}for
K.
2.4.1 (1)adbc#0; ( 2 ) a d  b c = 0 , a + d # 0 ; ( 3 ) a d  h c = a + d = 0 , a 2 + b 2 + c 2 + d2 # 0. cod(S,) = 0; cod(S,) = 1 ;cod(&) = 2. Linear vector fields satisfying (2.4.1) and (2.4.3) have codimension 1 and 2, respectively.
2.4.3 (i) A = t ( b + j ) , B = c , C=O,D=e/2 , E = j , F = O ; a = a t f . f i = d . 2
(ii) A = t ( b + j ) , B=c , C = 0 , D = a, E = j , F=O; y = d , 6 = e + 2 a .
2.4.5 Consider the types of Jordan block which give rise to nonhyperbolic linear systems. Show that each type of block satisfies a resonance condition for all r 2 2.
2.5.2 Observe that c > 0 implies f(x) > (< )  x for x suficiently small and positive (negative).
2.5.3 Use (2.5.8) for complex form with n = 2. Observe that J,,, = 0 for m, = 0 implies no %dependent terms arise. Alternatively, use (2.5.8) with 11 = 1 and a single (complex) variable z (see Exercise 2.5.2). Note Lq+' = A.
2.5.4 a = exp(3ia)/[l  exp(4ia)l. Note exp(4ia) # I for a # Znplq, q = 1, 2, 3,4.
398 Hints for exercises Hints for exercises
2.6.1 M = 3 has eigenvalues *i but
2.6.2 Note: (i) if AB = BA then exp(A + B) = exp(A) exp(B); (ii) N" = 0.
2.6.3 Let S 'MS= J, find In J from Exercises 2.6.1 and 2. exp(ln J ) = J implies L = S I n J S  ' .
Alternative implies state transition matrix (itself a particular fundamental matrix) (p(t, 0) = U(t) exp(Ct), make change of variable x = U(t)y and show that y = Cy. Thus alternative implies Theorem 2.6.1 with A = C and B(t) = U(t).
Theorem 2.6.1 implies x = A(t)x has solutions x(t) = B(t) exp(At)yo = ~ ( t , O)xo = Q(t)Q'(O)x, for any fundamental matrix Q(t). Let yo = Q'(0)xo to obtain the statement given in the question and yo = xo for B(t) = cp(t, 0) exp( At).
2.6.4 If Jc is a Jordan block corresponding to a complex eigenvalue L2 of M2 then the complex linear transformation that reduces Jc to the real Jordan form J,, transforms In Jc into a real matrix, i.e. JR has a real logarithm.
If P = (p(2n, 0) then P2 = ~ ( 4 n , 0 ) = exp(4nA), for real A, by the first part of the question. Show that B(t) = ~ ( t , 0) exp( At) is 4xperiodic in t.
cash 2n sinh 2')
{ (0 I)} (p(2n, 0) = =exp 2n
sinh 2n cosh 2n 1 0
2.6.8 (a) z l ~ ( ~ , z1zI4: (b) zlz12, zlzI4, I4 exp(2it).
2.6.9 For ,Ii = 0, i = 1,2, (2.6.14) implies resonance only if 11 = 0, i.e. all timedependent
y2 . terms can be removed. Let x = y + stn(2t) and find a = 4, b =  c . 4
0  P 2.7.1 A1 Ec given by (a)(l O ) ; (b)(: A); :). (b)gives unbounded motion.
2.7.2 Decompose R" into the direct sum F 8 Ec @ E" and consider restrictions of exp(At) to Es and Eu.
2.7.3 C", unstable.
2.7.4 No, origin is hyperbolic node. Maximum differentiability given by [bla]. Y1
2.7.5 aZj = 0, a,j = 0, = 4j, aSj r 0, aGj = 2(6jt ')(I  (SFf I ) ; 1 converges for i = 4 i f p < $ and for i = 6 i f P < Q . i=o
2.7.6 For C #O centre manifold is nonanalytic. '0
2.7.7 Assume centre manifold given by y = 1 nixi and show that a, = a , = 0; i = o
a,, = (k  I)!, a,, + , =,O, k 2 I . Hence y has zero radius of convergence.
2.7.8 Assume centre manifold Ec of the form y =a, + a ,x + a2x2 + 0(x3). Show that a, = a , = 0, a, = 1. Consider restriction of system to EC.
2.7.9 Assume centre manifold of the form x = co + c,y + cZy" 0(y3) and show that c0=c1 =o, c 2 = a.
2.7.10 r = 1, linearise; r > 1, assume Ec of the form y = a,x2 + 0(x3).
2.8.1 Polar blowingup gives: (i) saddles at 0 = 0, n, i > (< ) 0 for r > 0 and 0 = 0 (n); (ii) 0 = 0 unstable node, 0 = n stable node.
2.8.2 Singularities on r = 0 circle are: (a) 0 = 0 unstable node, 0 = n/4 saddle, 0 = n/2 unstable node, 0 = a stable node, 0 = 5x14 saddle, 0 = 3n/2 stable node; (b) 0 = 0 unstable node, 0 = n/4, n/2 saddles, 0 = n stable node, D = 544, 3'12 saddles.
2.8.3 Repeated blowingupalongpositive yaxisgivesfurthersaddlenodesingularities.
2.8.4 Division by lulk and Ivp is necessary to prevent orientation reversal.
2.8.5 Positive xblowup, unstable node; negative xblowup, stable node. (cf. Exercise 2.8.1 with a =  I , b = 2.)
2.8.6 (a) Do polar blowup, investigate resulting singularities at O=n/2, 3x12 with further polar blowups. Obtain nonhyperbolic saddle.
(b) Polar blowup gives six hyperbolic singularities. Obtain 'monkey' saddle.
Note that the unfoldings of the vector fields considered in this question appear in Section 5.6 (see (5.6.2) (q= 2) and (5.6.14) (q = 3)). The reader may like to confirm that the underlying singularity for q = 5 (see (5.6.34) and (5.6.35)) is a focus, while q = 4 (see (5.6.21)) admits a variety of singularity types.
Chapter 3
3.1.1 Recall: (i) the spectral radius, p(A), of A is the maximum of the absolute values of the eigenvalues of A; (ii) the spectral norm, o(A), of A is the positive square root of the
largesteigenvalue ofATA; (iii)p(A) ,< a(A); (iv) a(A) < JlAJ(,where J(A1 = 1 (aij(. ij
(a) Let M'AM = D, D = [AiSi,]. Consider det(M 'BM  pl), with I( an eigenvalue of B that is not equal to Li for any i = I , . . ., n, and show that o(D;'C,) 2 1, where D, = D  p1 and C, = M'CM. Observe
max[(li  pJ '1 c ' implies min[Ji,  pi] < c. i i
0 j # i + l (b) Let M 'AM = D + T, T = [till, tij = l o r 0 j = i + l . Proceed as in (a) and
n  l I
note that ( I + D i l T )  ' = I + 1 (1fDikTk, o(T)= 1, 1 g < n x " for k = l k = l
x > l . (c) Observe similar results follow for &(A).
3.1.2 Hyperbolic implies structurally stable: use Exercise 3.1.1. Structural stability implies hyperbolic: observe a Jordan block J associated with an eigenvalue I. of absolute value unity satisfies (Jkx( = 1x1 for all k~ Zt if x lies in the eigenspace of A. The block (1  6)J, S > 0, can be made arbitrarily close to J but lJkxl , 0 as k t co for all x. Density: note that, if A is nonhyperbolic with eigenvalues Li, then A + 61 has eigenvalues 1, + S.
3.1.3 A structurally stable in S: let A, B E S be &close and apply Theorem 2.1.3 in the
Hints jor exercises Hints for exercises 401
subspaces on which the restriction of A and B is not the identity. Hence construct a coniugacy for A and R. A is not structurally stable in L(R2): consider
Restriction of A to its stable manifold is orientationreversing while for (3.4.12) it is orientationpreserving.
f ( P , : (x, y)w(5x  2, yl5  215). Show that A€O(R2) is a rotation so that every circle, centre x =0 , is invariant under A. Observe B = (1  c)A,e 1 0 . has no invariant circles. Prove that conjugacy preserves invariant circles.
3.5.3 (a) Note g(Q  f; use (3.5.3) and (3.5.1). (b) Explicit form of f given in Exercise 3.5.1 shows xcomponent of f(x, y),
(x, ~ ) E P , u P , , is independent of y. Let X E 0 Q("'and X'EA have the same neN
(a) Use the Implicit Function Theorem and Exercise 3.1.1 (b) Use Exercise 3.1.1. (c) D%(o) = DX(0). xcoordinate, show that If"(x)  f"(x'] + 0 as n + a. Eliminate X E Q\ 0 Q'"'.
neN
Similarly, for X E n Q'"' using f  ' in place off . neZ '
(a) 1q1< 614; (b) 191 < 432; (c) 1~11 <F./13. Use Dq2,(r0)=exp(2nDX,(r0)) (see Exercise 2.2.2). where X, = i and X,(r,) = 0.
(a) 161 < ~ / ( 2 + R); (b) 161 < c/(R2 + 2R). The fixed point x* is not hyperbolic.
Theorem 3.3.1 is: (a) applicable. nonhyperbolic closed orbit; (b) applicable, nonhyperbolic fixed point at (x, y) = (I. 0); (c) not applicable, use perturbation (6B(y) , O ) , 6~ R.
3.5.5 Cf Propositions 3.5.35.
3.5.6 (b) Consider h: I: + Z defined by h(u), = o(,_ ,,. 3.5.7 Observe that ifaq(u) = u then u is periodic with periodq' where q'lq. 335 period12
orbits. (a) Theorem 3.3.3(i) fails; (b) Theorem 3.3.3(iii) fails.
3.5.9 A is repeated within itself on all scales. i = sin(2nx), j = 6 sin(2ny) (a) for 16) < J( l + 2n) is I:C'close to
i= l , j = 2 + 6 s i n [ 2 n ( y  2 ~ ) ] (b) for 161 < &/(I + 671). 3.5.10 Fixed points: ( 113, 113). (112,  112). Period2 points: (4/13,6/13), (6113, 4113).
Apply Theorem 3.3.1 to S,. The EC1close perturbation given by i = E + [r cos(2nr)/(l + r2)], 0 = 1 has no limit cycles for Irl> I/E.
I
3.5.1 1 (a) Verify for r ) gn(P,") and use induction. Exercise 3.5.9 gives: n = o
(i) square of side 215, centre (x, y ) = (2/5,  2/5); (ii) square of side 2/S2, centre (x, y) = (  8/25,  12/25); (iii) square of side 2/5', centre (I, y) = (38/125,38/125).
(b) u = ( . . . v  , N  ~ p . . . ,YN, ) I  ( N  I ) ~ . . 'Io.VI,. . . ~ V N , . . .). (c) Central block of 2N symbols must be preserved for k shifts of binary point to
the left, i.e. . . u = { . . . i ,..., i , i ,..., 1 . 1 ,..., i , . . . },i={O,l}.  
k N N
Maximum number of blocks that can be reached in kiterations is 2'. (d) 'Chaotic' motion.
Use appropriate lifts to examine fixed and periodic points off. (a) infinite number of periodic points implies nonwandering set not finite. (b) no fixed points but every orbit is dense therefore nonwandering set does not
consist of fixed and periodic points. (c) four fixed points on S1 but none are hyperbolic. (d) nonhyperbolic period2 points.
Recall that n: R2 + T 2 is a local dilfeomorphism and differentiate n@(x)) = fq(n(x)) w$h respect to x to show that TP(n(x)) (see $3.6) and Dp(x) are conjugate. DP(x) = A', for all x, and A' is hyperbolic. W""(P'(n(xS)) = n(Aix* + E'."), where
F(Eu) are the stable (unstable) eigenrpaces of A. W5.Y = bl WS."(P(n(x*)). i = O
3.5.12 Note that if (x, ~ ) E A is given by h(u), then x is determined by the part of u lying to the left of the binary point. If A, denotes the invariant Cantor set of/,, then show that (fl(x)/,(x1)(=5(xx'l for any X E A ~ , x1$AI, i.e. /, is locally repelling at each point of A,.
Observe that n(x*), where x* = (Aq  1)'p, p€Z2, is a periodic point of f of period at most q. Verify that, when A is given by (3.4.12). IDet(Aq [)I, q 2, is an integer greater than unity, hence show that (Aq  I) ' has at least one element that belongs to Q"Z2. When q is prime deduce that f has a periodq point.
Differentiable conjugacy off, g by h implies (&#(x))) = (g(k(x))) + k where k E Zn and ; denotes a lift of a . Differentiate with respect to x and set x = 0. Conditio~ls on C mean_it is a lift of a diffeomorphism, h, say, on T". CAx = BCx implies h#(x)) = g(h(x)); take projection n and show h(f(0)) = g(h(0)). where 0 = n(x).
3.5.13 Show that a, and o2 both represent the point (f, 0). The map h fails to be injective at points corresponding to the dyadic fractions (ml/2"1, m2/2"'), mi, niE Z +, i = 1,2 (see Arnold & Avez, 1968, p. 125). If these points are disregarded, the symbolic dynamics can be used to show that the periodic orbits are dense in T2.
3.6.1 j(0) = (1, (g:y)(O) = (2, O)T.
3.6.2 Overlap map h12(x,)=4/x,. (b) For U E TU2,,(=R), Ilullx, =4lvl/x: on U2\P2, where 1.1 is the Euclidean norm of . .
v = ((I + 13"2)/6).x. Irrational slope implies stable and unstable manifolds of fixed point at x = O wind densely around the torus without closing. Homoclinic point is given by the intersection of y = ( I + 131'2)x/6 and y = (I  13'!2)(x  1)/6.
402 Hints for exercises Hints for exercises 403
Let (Ui, hi), i = 1,2, be overlapping charts and assume that h,, = h,h; I : U , . U , is C'. Show that D(h,fh;')(h,(x)) and D(h,fh;')(h,(x)) are similar. If f(x*) = x* the eigenvalues of the tangent map Tf,. can be unambiguously defined to be those of its local representatives. This means that a fixed point on M is hyperbolic if all its local representatives are hyperbolic in the sense of Definition 2.2.1.
Chart T 2 with I and define /TC(vo)ll = ID~(X)V,), v,e TT: and v, = Tnilv,. A has eigenvalues 1, = (3 + SLI2)/2, i2 = (3  S1")/2. Take P = IL,I = Ii,I, C = 2 and c = t
(a) f, is an Anosov automorphism; Theorem 3.6.1 applies; T2 connected implies there is only one basic set 0, = T2.
(b) f, has no periodic points but R = T2 (note f:"(x, y) = (x, y + 2n(3'I2)) mod 1, neE); Theorem 3.6.1 is not satisfied.
(c) Four hyperbolic fixed points, PI , . . ., P4; Theorem 3.6.1 applies; f l =
{PI, P2, p3, P41.
Recall: a Cantor set is a closed, uncountable set with empty interior such that every point is an accumulation point.
Note (f:r)(O) points into the image of y under f and Au A Av = Det(A)(u A v).
X* = h(u*), u* = (. . . :u'q' i .u"' i i . . .). Let xt = h(ot) where
where dq)= a,u2 and o, is a subblock of d9' containing q  k symbols. Show that aY(ut) + u* and a"q(ut) + $(u*) as n , co. No.
(E3.2) is an autonomous system.
Transverse heteroclinic points for b/a < i n o sech(no/2).
Chapter 4
4.1.1 (a) Terms of order r 2 3 are not removed as in (4.1.14). (b) Linear terms are not removed when p , #O. Transformed system is not an
unfolding of j =  y2.
4.1.2 (a) Take X(p, x) nonversa1 and Y(v,x) versal. Show X  Y but Y *X. For example X(p, x) = x2, Y(v, y) = v + y2.
(b) (i) X . Y: h@,, k,. x) = x  p1/2, cpko, p1 = jio  ~ 3 4 ; Y  X: h(v,, X) = y, ~ ( v , ) = (v,, 0).
(ii) Let x = x*(p,, p,) be fixed point o f f = Xk,, p,, x). Then X  Y: h b , j t , ;x)=xx*(j~~,p,) , Q ~ O , /41)= (PI  ~x*@o, PI)', 3~*@o, PI));
"I YX: h(v0, V,, y)=y, Q(v,, v,)= 3
4.1.3 Let x=ay , then I= Y(q, y), Y(0, y)=ay2, becomes i= X(q,x)=aY(q,a'x) so
that X(0, X) = x2. Note for a < 0, x = ay is an order reversing homeomorphism of R. Use venality of i = 10 + x2 and transform back.
x2 + p2x + p 4.1.5 (a) q(p, x) =    is not continuous at (11, x) L (0,O) for any choice
px3 + x2 + j12x + p of q(O.0).
(b) Comparison of coeficients yields h,(p)= I or h,(p)= p2. Latter implies sl@) not defined at p = 0 and therefore certainly not smooth on neighbourhood of p = 0. Hence b,(y) E I, q(ji, X) = (1 + ~ I X )  I, S, (p) E 0, so@) = p.
q k , x) (c) Let 4(p, X) = ,SO(p) = sO(p), whereq(p, x) and s,(/c)are given in (E4.7).
(1 + 1.x)
4.1.6 (a) Take G(p, .v) = xk in Mather Division Theorem. Set p = 0, differentiate k times and conclude Q(0,O) = l/q(O, 0) = g(0).
(b) Let X(p, X ) be any unfolding of i = xk and take F = X in (E4.14). Since Q(0,O) =  1, X(p , x) is equivalent to family induced by (E4.15) with q(p) =
(so(p), . . ., st ,(pHT. "k1 .
(i) Let y = x   In (E4.15). k
(ii) For k odd, right hand side of (E4.15) has at least one real zero. 4.2.1 (a) (iv), (v); isoclines arc tangent to each other but neither is tangent to either
coordinate axis. (b) (iv) and (v); (v).
4.2.2 If j = 0 then 2yx2  x  (2y  2y3  1) = 0.Take y # 0 and examine the discriminant ofthisquadraticequation. Equation (E4.19) arises in connection with the averaged forced Van der Pol oscillator (see Arrowsmith & Place, 1984).
4.2.3 (a) Observe distance, d(p, x), between i = 0 and j = 0 isoclines satisfies d(0, x) = x2 and use Malgrange Preparation Theorem;
(b) (Y  x3 + PX, yIT; (c) (y  x4 + p.x3, ) I ) ~ , (y  x4 + 3px2  2ji2, y)T, for hyperbolic points isoclines
must intersect transversely  three such intersections are not possible for d(0, x) = x4.
4.2.4 Show that +(x) has a unique maximum on (0.11 at x = j.
4.2.5 (a) P = 0 isocline is given by an increasing function of H bounded by y with slope y / / l at H = 0. The H = 0 isocline is independentpfll and 1. A single fixed point arises at tangential intersection of H = 0 and P = 0 isoclines.
(b) Find where the nontrivial 6 = 0 and p = 0 isoclines meet in a single point. More details of both of these models appear in Arrowsmith & Place (1984).
4.2.6 (a) Evaluate LAxT1x;'ei, i = 1, 2, where ei is the ith column of I,,, and A is the coeficient matrix of the linear part of the extended vector field in (4.2.16). Use (2.3.7).
(b) Consider L,x,pke, and LAx,pke,. (c) Apply Theorem 2.7.2 to the 2jet of the transformed extended system to obtain
equivalent system with i, =Ax,. In absence of terms of order three and higher, i , depends only on x, and p. Complete the square on s, to obtain (4.2.2)
(Q(P))~ with a, = 0, b, = c,,, v = P(p)  , where 4 C 2 3
412 Hints for exercises
6.8.2 (a) Use repeated root of (6.8.5) in (6.8.8) and expand in powers of v,. (b) Use the Implicit Function Theorem on (6.8.8). Substitute r = r(w, v)(l + a ) in (6.8.4).
6.8.4 introduce metric distance between two circles as maximum radial displacement. Show that associated functional equation which maps circles using N, is effectively the second component of (6.8.12).
6.8.5 Use induction on k and show that ]Nk(z) + exp(2nipk/q)cifq'I2 = JNk(z)J2 up to order lzIq.
6.8.6 Express in polar coordinates, take logarithms and separate real and imaginary parts. Introduce local coordinate r, and consider Taylor expansion.
6.8.7 Use Theorem 5.4.2 and (5.4.14). Resonance tongue with tip at (0, i,). Show that ( v , , v , ) near (0, V,) and (Re I , Im I ) near (cos(2np'/q'), sin(2np'/q1)) are related by a local diffeomorphism (cf Figures 5.5 and 5.6).
6.8.8 Use generalisation of (6.8.5) and (6.8.8).
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actionangle variables, 48 additional resonant terms
areapreserving planar map, 308 rational rotation, 86, 258 timedependent vector field, 91
Anosov automorphism and Anosov diffeomorphisms, 134 chaotic basic set, 159 definition, 133 and dynamical systems, 120 homoclinic points, 1368. 182 periodic points. 134
Anosov diffeomorphism conjugacy to automorphism. 134 and dynamical systems, 133 structural stability, 135
areapreserving planar map Birkhoff normal lorm, 305 complex lorn, 306 and Htnon areapreserving map, 169 normal lorm, 308
Arnold's circle map, 248 and dissipative standard map, 349 and unloldings 01 rotations, 257
Arnold tongues delinition, 252 symmetry 01, 293 and unloldings of rotations, 257
atlas. 2 attracting set, 346
chaotic. 161 attractor, 346
strange, 161 AubryMather Theorem, 338 axiomA diffeomorphism, 158
Baker's translormation, 184 basic sets
Anosov automorphism. 159 Decomposition Theorem, 1589 horseshoe diffeomorphism. 158 spinning dilliomorphism, 160
INDEX
bifurcation, local, 191 bilurcation curve, 207 bifurcation point, 190 biharmonic oscillator. 48 Birkhoff attractor, 348 Birkhoff normal lorm
for areapreserving planar map, 305 for Hamiltonian, 303
Birkhoff periodic orbit, 336 Birkhoff periodic points, 309
01 type (p,q), 336
Birkhoff rotation set, interval, 347 347 Birkhoff Theorem, 338 blowingup
for cusp singularity, 107 directional, 105 polar, 102 in in xdirection, ydirection. 106 106
Bogdanov map, 359 Bogdanov points, 378
Calabi invariant, 345 C1norm lor vector field, 123, 125 canonical polar coordinate system, 305 canonical translormation, 45 Cantor set
definition, 333, 386 in double invariant circle bilurcation, 375 invariant
lor areapreserving twist homeomorphism, 338
lor circle homeomorphism, 3325 for horseshoe diffeomorphism, 13947;
dynamics on. 14954, 184 in SmaleBirkhoB Theorem, 165 for spinning dikomorphism, I 6 0
centre eigenspace, 94 centre manilold
differentiability of, 97