Nonlinear Dynamics & Chaos MAT3007 2010 D J B Lloyd
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Nonlinear Dynamics & Chaos MAT3007 2010 D J B Lloyd Department of Mathematics University of Surrey March 21, 2010 1
Transcript of Nonlinear Dynamics & Chaos MAT3007 2010 D J B Lloyd
Nonlinear Dynamics & Chaos
MAT3007
2010
D J B Lloyd
Department of MathematicsUniversity of Surrey
March 21, 2010
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Contents
1 Introduction 4
2 Mathematical Models 10
3 Key types of Solutions 13
3.1 Steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Chaotic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Phase portraits and topological equivalence: flows 17
4.1 1D Phase portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 2D Phase portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 Stability and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Stable and Unstable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Bifurcations: flows 25
5.1 Saddle-Node Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Transcritical bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Pitchfork bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.5 Global bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.5.1 Homoclinic bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.5.2 Period doubling route to chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 One-dimensional maps 42
6.1 Cobwebs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Topological conjugacy and linearisation . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Period doubling bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.4 Periodic windows and Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.5 Period 3 Implies Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.6 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.7 Universality and Re-normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 Chaotic 1D maps 63
7.1 The Doubling map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.2 The Tent map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.3 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8 Fractals 67
8.1 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.2 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9 Strange Attractors and Repellers 73
9.1 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 2D maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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10 Application: Secret communication with chaos 77
11 Matlab codes 80
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1 Introduction
The study of chaotic systems and fractals is a relatively new mathematics topic. A long, long timeago, we had no means to predict the future, understand the laws of motion, explain what lead certainnatural events. Over centuries, several physicists, chemists, biologist and mathematicians started tounravel the laws of nature and develop techniques that allowed us to predict the future. In particular,a certain Issac Newton developed calculus and started applying it to all sorts of problems from themotion of the planets, tides, mechanics and was able for the first time to predict accurately whatwould happen. Over time, science found out that nature could be predicted, analysed, recorded andexploited for our benefit. By the 18th century, scientists had been so successful in understandingnature that many thought that there was nothing left to discover.
The laws of nature were written down as dynamical systems describing the motion of every particlein the universe, exactly and forever. The goal of the scientist was to examine the implications ofthe laws for any particular phenomena of interest. As Ian Stewart (Professor of Mathematics at theUniversity of Warwick) wrote in his excellent book Does God play dice? [4], “chaos gave way to aclockwork world”. However, it turns out not to be that easily predicting the future. . .
Course text book: Both the lectures and notes are strongly based on the excellent course notesof Prof. Bernd Krauskopf, Dr. Hinke Onsinga [2] and Prof. John Hogan [1] at the University ofBristol and the excellent text book by Steven Strogatz, Nonlinear Dynamics and Chaos, publishedby Westview [5]. Exercises and questions are frequently taken from the text book where some hintsand answers can be found at the back of the book. I strongly recommend getting a copy of thisbook, though not essential, will help (I don’t get any royalties :-)
Matlab codes and Movies: All the computations in these notes are carried out using Matlab.The codes for all the computations can be found on the Nonlinear Dynamics & Chaos website
[http://personal.maths.surrey.ac.uk/st/D.J.Lloyd/MAT3007.php]
It is hoped that students will use these codes to explore the phenomena discussed in lectures. BothMatlab codes and movies demonstrating chaotic phenomena will also be presented in lectures. Atthe end of these notes is a listing of Matlab codes and how to use them in §11.
Hyper-linked notes: Considerable effort has gone into hyper-linking these pdf notes. This meansthat if you have an internet connection, clicking on the web-links will take your web browser there.Furthermore, sections, equations, figures, examples, theorems etc. are also hyper-linked allowingyou to easily move through the pdf document.
I hope you enjoy this course!
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Outline of the lectures:
1. Outline of module, examples of dynamical systems and observations of chaos: chaotic water-wheel and chaotic double pendulum.
2. Definition of dynamical systems and topological equivalence.
3. Phase portraits, linearisation and classification of equilibria
4. Linearisation of nonlinear flows, stable and unstable manifolds
5. Examples class going over exercise sheet 1
6. Introduction to bifurcations, saddle-node bifurcation
7. Pitchfork bifurcation and Hopf bifurcation
8. Hopf bifurcation analysis and chemical oscillations
9. Global bifurcations: homoclinic bifurcation
10. Global bifurcations: period doubling route to chaos. The Rossler system.
11. Poincare sections, period doubling bifurcation and Lorenz maps.
12. Examples class going over exercise sheet 2
13. Introduction to 1D maps
14. Linearisation, phase portraits, cobwebs, topological conjugacy
15. Saddle-node and transcritical bifurcation in maps
16. Period doubling bifurcation in maps
17. Periodic windows and intermittency
18. Lyapunov exponents
19. Introduction to renormalisation
20. Renormalisation
21. Examples class going over exercise sheet 3
22. introduction to chaotic 1D maps. Symbolic dynamics, shift map
23. Introduction to Fractals, Cantor set and Kock curve
24. Similarity and box dimension
25. Strange attractors, strangle saddles
26. Examples class going over exercise sheet 4
27. Application: Secret communication with chaos. Movie
28. Examples class going over worksheet 5
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Some examples of dynamical systems:
Example 1: Newton’s cooling
One of the first things Newton described using calculus was the cooling of a liquid. Experimentally,he found that the rate of cooling was directly proportional to the temperature difference betweenthe object and the surrounding background. In terms of a dynamical system, Newton’s cooling lawcan be written as
dx
dt= −α(x− xs), (1)
where x = x(t) is the temperature of the liquid, α > 0 is the constant of proportionality and describesthe “material” properties of the liquid (water will cool down at a different rate to say honey). xs isthe background air temperature (e.g., 20 Celsius).
If we know the initial temperature of the liquid, then (1) has a unique solution for all time!
Let us take the initial temperature x(0) = x0 and solving the ODE (1) yields
x(t) = (x0 − xs)e−αt + xs = Φt(x0),
where Φt is known as the evolution operator. Given any x0, xs and a value of t, we can find a uniquesolution. The evolution operator defines a flow starting from x0. Note that the equilibrium (steady)state given by dx/dt = 0 is x = xs. The solution x(t)→ xs as t→∞ since e−αt → 0.
If we take x0 = 50, xs = 25 and α = 0.5 we get the following solution
0 1 2 3 4 520
25
30
35
40
45
50
t
x(t)
x(t) = Φt(x0)
xs
Figure 1: Solution of Newton’s cooling (1) for x0 = 50, xs = 25 and α = 0.5. Note the solution is unique!i.e., there is only one curve emanating from x(0) = 50.
Example 2: Population dynamics
There have been many laboratory studies of the population dynamics of single species e.g., flies,worms, ants etc. Thomas Malthus 1798, was the first to write down some rules for the evolution ofthe size of populations. A natural law to write down for the rate of change of the population is
dN
dt= births – deaths + migration, (2)
where N(t) is the size of the population at time t. This equation is called the conservation equationfor the population.
The simplest possible model has no migration (for example in a closed lab) and the births and deathsare proportional to the current population size N(t) e.g.,
dN
dt= bN − dN ⇒ N(t) = N0e
(b−d)t = Φt(N0), (3)
where b, d are positive constants and the initial population is N(0) = N0. Again we can write downthe evolution operator Φt = e(b−d)t that describes the evolution (flow) from N0 to N(t) for somegiven t.
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From (3), we find that the population grows exponentially if b > d or dies out. Not particularlyrealistic since one would expect there to be some sort of self-limiting process stopping the exponentialgrowth when the population becomes too large.
A more realistic model is if the number of deaths increased if the population is too large e.g.,d = fN(t). Now equation (2) becomes
dN
dt= bN − fN2 = rN(1−N/K), (4)
where now the per capita birth rate is r(1−N/K) and K is the carrying capacity of the environmentthat is determined by the available sustainable resources and r,K > 0. We have also set b = r andf = r/K.
As for Newton’s cooling, we can calculate the steady steady states by setting dN/dt = 0 to findN = Ns = 0 (i.e, no population) and N = Ns = K (the population have reached the carryingcapacity of the environment). What happens if we have a population whose initial size is close toeither N(0) = 0 or N(0) = K?
To do this we look at the the evolution of a small perturbation N(t) where |N(t)| � 1 (very muchless than 1)
N(t) = Ns + N(t) (5)
where Ns = 0 or Ns = K are the steady states. Now if we start off with N(0) small, then if N(t)→ 0the population size N(t) → Ns and we call Ns a stable steady state. On the other hand, if N(t)becomes large, then the population size N(t) does not stay close to Ns and we call Ns an unstablesteady state.
We can find out what happens to N(t) by substituting (5) into equation (4) to find
d[Ns + N(t)]
dt= r[Ns + N(t)](1− [Ns + N(t)]/K),
dNsdt
+dN
dt= rNs + rN − r
K(N2
s + 2NsN + N2),
= rNs(1−Ns/K) + rN(1− 2Ns/K − N/K).
Now, we know that dNs/dt = 0 since it is an equilibrium (by definition) and also rNs(1−Ns/K) = 0since Ns solves the right hand side of equation (4) equal to zero. So we have
dN
dt= rN(1− 2Ns/K − N/K).
But, if N is very small then N2 is even smaller and we can ignore the N2 term to get
dN
dt≈ rN(1− 2Ns/K)
If we set Ns = 0 in this equation, we find N(t) ≈ Cert. Since r > 0, ert → ∞ as t → ∞ andhence N(t) becomes very large. Therefore Ns = 0 is an unstable steady state. On the other hand, ifNs = K, we find N(t) ≈ Ce−rt → 0 as t → ∞. Hence, N(t) → 0 and so Ns = K is a stable steadystate.
In other words, if we start the population near the carrying capacity of the environment, then thepopulation size will tend towards to carrying capacity whereas if we start with a small perturbationthen we will no longer remain small.
We can actually say more than this for equation (4). Using the method of Separation of variables,we can solve (4) explicitly for any initial starting population size N(0) = N0
N(t) =N0Ke
rt
[K +N0(ert − 1)]= Φt(N0),
where again we have defined the evolution operator Φt that acts on a given starting population sizeto yield a population size for a given value of t.
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26 BioScience • January 2001 / Vol. 51 No. 1
Articles
Reproductive and survival changesSnowshoe hares can have three or four litters over a summer,with five leverets on average in each litter. All hares begin tobreed in spring when they are 1 year old, so age at sexual ma-turity is a constant. If reproductive rates are to vary, only lit-ter size or the number of litters can change. Lloyd Keith andhis students at the University of Wisconsin, working in cen-tral Alberta, supplied the first detailed description of the wayin which hares change their reproductive rate over a 10-yearcycle (Cary and Keith 1979). Reproductive output reachesits peak very early in the phase of population increase, whenfemales are producing 16–18 young per summer. It then be-gins to fall rapidly while numbers are still rising; it reachesa nadir in the year of the density peak or 1–2 years thereafter,during the decline phase of the cycle. The response of re-productive output to hare density seems to lag 1–2 years, sothat, as shown in Figure 2, for example, the lowest repro-
ductive output occurred in 1973, 2 years after the densitypeak of 1971. We found a similar but not identical patternof response in hares in the southwestern Yukon. The re-productive rate at the cyclic peak in the Yukon was about two-thirds of the maximum, compared with one-half of themaximum in the Alberta peak. But reproduction in both ar-eas was minimal in the decline phase (about one-third ofmaximum reproduction) and highest in the early increasephase of the 10-year cycle (Stefan 1998, Hodges 2000).
Changes in mortality rates are the other driver of changesin hare numbers over the cycle. At Kluane Lake in the Yukonwe measured survival rates of hares with radio collars dur-ing two population cycles. Additional data on survival ofhares over two cycles in central Alberta was obtained throughmark-and-recapture methods and radiotelemetry (Keith1990). The pattern of change for adult hare survival isshown in Figure 3. Adult survival rates begin to drop slowly
Figure 1. Canada lynx fur returns from the Northern Department of the Hudson’s Bay Company from 1821 to 1910. TheNorthern Department occupied most of western Canada. The cycle for these data averages 9.6 years. Data are from Eltonand Nicholson (1942). Photo: Mark O’Donoghue.
Figure 2. Changes in the annual reproductive output of female snowshoe hares in the Rochester area of central Alberta,1962–1976. Reproductive output was measured in autopsy samples. Data from Cary and Keith (1979). Photo: Alice Kenney.
26 BioScience • January 2001 / Vol. 51 No. 1
Articles
Reproductive and survival changesSnowshoe hares can have three or four litters over a summer,with five leverets on average in each litter. All hares begin tobreed in spring when they are 1 year old, so age at sexual ma-turity is a constant. If reproductive rates are to vary, only lit-ter size or the number of litters can change. Lloyd Keith andhis students at the University of Wisconsin, working in cen-tral Alberta, supplied the first detailed description of the wayin which hares change their reproductive rate over a 10-yearcycle (Cary and Keith 1979). Reproductive output reachesits peak very early in the phase of population increase, whenfemales are producing 16–18 young per summer. It then be-gins to fall rapidly while numbers are still rising; it reachesa nadir in the year of the density peak or 1–2 years thereafter,during the decline phase of the cycle. The response of re-productive output to hare density seems to lag 1–2 years, sothat, as shown in Figure 2, for example, the lowest repro-
ductive output occurred in 1973, 2 years after the densitypeak of 1971. We found a similar but not identical patternof response in hares in the southwestern Yukon. The re-productive rate at the cyclic peak in the Yukon was about two-thirds of the maximum, compared with one-half of themaximum in the Alberta peak. But reproduction in both ar-eas was minimal in the decline phase (about one-third ofmaximum reproduction) and highest in the early increasephase of the 10-year cycle (Stefan 1998, Hodges 2000).
Changes in mortality rates are the other driver of changesin hare numbers over the cycle. At Kluane Lake in the Yukonwe measured survival rates of hares with radio collars dur-ing two population cycles. Additional data on survival ofhares over two cycles in central Alberta was obtained throughmark-and-recapture methods and radiotelemetry (Keith1990). The pattern of change for adult hare survival isshown in Figure 3. Adult survival rates begin to drop slowly
Figure 1. Canada lynx fur returns from the Northern Department of the Hudson’s Bay Company from 1821 to 1910. TheNorthern Department occupied most of western Canada. The cycle for these data averages 9.6 years. Data are from Eltonand Nicholson (1942). Photo: Mark O’Donoghue.
Figure 2. Changes in the annual reproductive output of female snowshoe hares in the Rochester area of central Alberta,1962–1976. Reproductive output was measured in autopsy samples. Data from Cary and Keith (1979). Photo: Alice Kenney.
0 20 40 60 80 100 120 140 160 180 200
t
N(t)
K
N0
N0
N0
N(t) = Φt(N0)
Figure 2: Logistic population growth for equation (4). We plot the evolution of several population sizes withdifferent N0.
• You should check that N(t)→ K as t→∞!
For the interested student, the book by Murray, Mathematical Biology [3] is an excellent introductionto the subject of mathematical biology.
Summary so far:
• From Calculus we know how to solve systems of linear ODEs and even some nonlinear ODEslike equation (4)
• For any given initial condition (x(0) or N(0)) the solution to these equations are unique,smooth and well behaved.
However, the logistic equation (4) is one of the very few nonlinear equations that can be solvedexplicitly. Most of the time this is not possible! What happens when we can’t solve explicitly?
Example 3: Discrete Population Dynamics Ok, so we know how to solve the logistic equa-tion (4) and everything is nice, but the population size isn’t continuously changing. What happenswhen we make the change in population size discrete?
To do this, we will re-do Example 2, but replace dN/dt by the finite difference Nn+1 −Nn (similarto doing a numerical approximation of dN/dt).
Again, a natural law to write down for the rate of change of the population is
Nn+1 −Nn = births – deaths + migration, (6)
where Nn is the size of the population at time/step n. This equation is called the discrete conser-vation equation for the population.
The simplest possible model has no migration (for example in a closed lab) and the births and deathsare proportional to the current population size Nn e.g.,
Nn+1 −Nn = bNn − dNn ⇒ Nn = N0(1 + b− d)n = Φn(N0), (7)
where b, d are positive constants and the initial population is N0. Again we can write down theevolution operator Φn = (1 + b − d)n that describes the evolution (map) from N0 to Nn for somegiven n.
From (7), we find that the population grows factorially if |1 + b− d| > 1 otherwise it dies out. Notparticularly realistic, as before, since one would expect there to be some sort of self-limiting processstopping the exponential growth when the population becomes too large.
A more realistic model is if the number of deaths increased if the population is too large e.g.,d = fNn. Now equation (6) becomes
Nn+1 −Nn = bNn − fN2n ⇒ Nn+1 = Nn
[1 + r − r
KNn
], (8)
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where b = r and f = r/K.
We may do a little rescaling with Nn = [(1 + r)/r]Kxn and setting 1 + r = r we get the famouslogistic map
xn+1 = rxn(1− xn). (9)
Now, unlike the differential equation version of (9) or (8) there is no explicit general solution! Sowhat happens? Surely, the dynamics are the same as those for (4) where all the solutions convergedto the steady state N = K?
Well lets run a Matlab program
r = 4
x(1) = 0.2
for n = 2:50
x(n) = r*x(n-1)*(1-x(n-1));
end
plot(x);
Running this yields the following diagram
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n
xn
Figure 3: Chaotic rabbits? Plot of Matlab computation of xn+1 = 4xn(1 − xn) with x1 = 0.2. Note thatunlike the logistic differential equation (4) there is no convergence to any steady state! Matlab code for figurecan be found here: [Chaotic Rabbits]
So by making the population dynamics discrete instead of continuous, we have got some very com-plicated dynamics; sometimes called chaos. Is this typical? Does it only occur for maps? How canwe analyse it?
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2 Mathematical Models
Mathematics becomes really powerful when we try to predict the future. For this entire course, wewill be solely interested in mathematical models that describe how a (continuous) quantity(s) evolvewith time via a set of rules (e.g., Newton’s rule for the rate of liquid cooling (1) or rate of change ofpopulation dynamics (2)). These types of mathematical models are called Dynamical Systems.
Definition 1 (Dynamical System) A dynamical system is a system whose behaviour can be de-scribed by an evolution operator
Φt : X → X,
defined on a space X for all t ∈ T . The space X is called the state space or phase space. The spaceT , or time, can be R i.e., time is continuous or T = Z in which case time is discrete. The evolutionoperator must be such that the following two conditions hold for any initial condition x0 ∈ X andany t, s ∈ T :
1. Φ0(x0) = x0 (“no time, no evolution”),
2. Φt+s(x0) = Φt(Φs(x0)) (“determinism”).
If T = R, then the dynamical system is a continuous time system, which is usually given as a systemof differential equations (e.g. liquid cooling (1))
x = f(x), (10)
where f is a function defined on X. The evolution operator Φt described the flow of the vector field.
If T = Z then the dynamical system is a discrete system and is defined by a map (e.g., populationdynamics (6))
x 7→ g(x), (11)
where g is a function defined on X. The evolution operator is the map g itself and Φt(x) = Φn(x) =gn(x).
We can find out what happens to the initial point x0 as time evolves by applying the evolutionoperator Φt. As we increase time, we could follow x0 under the flow of (10) or compute the iteratesof x0 by applying the function g in (11). If f in (10) is sufficiently smooth, then we can go back intime and follow the flow backward to see where x0 came from. The entire future and history of apoint x0,
{Φt(x0) | t ∈ T},is called the orbit of x0. For discrete time systems, g must be invertible in order to look at the entireorbit. In this case we will only consider the forward orbit {Φt(x0) | t ∈ T, t ≥ 0}.
Remark 1 Throughout this course we will always assume that the function f in (10) is sufficientlysmooth so that both the past and future are uniquely determined by the initial condition x0 (see 2ndyear Ordinary Differential Equations course).
Example 4: The discrete linear population model (7)
Nn+1 = (1 + b− d)Nn.
This is a discrete dynamical system for any b, d ∈ R. We may write this as a map
N 7→ (1 + b− d)N = g(N)
with state space X = R and evolution operator Φt(N) = Φn(N) = gn(N). Orbits are given byiterates of the map. Note this map is invertible since for any given Nn+1 point we can find only oneNn point
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nn
Nn+1g(x)
Figure 4: Figure showing the discrete linear population model (7) is invertible i.e., for a given Nn+1 iterate,there is only one Nn point that is came from.
Example 5: The logistic equation
xn+1 = rxn(1− xn),
is a discrete dynamical system for any r ∈ R. We may rewrite this as a map
x 7→ rx(1− x) = g(x),
with state space X = R and evolution operator given by Φt(x) = Φn(x) = gn(x). Orbits are givenby iterates of the map.
Note this map is not invertible since for a given xn+1 iterate there may be two xn iterates that itcould have come from.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
xn
xn+1
? ?
g(x)
Figure 5: Figure showing the non-invertible nature of the logistic map. For a given xn+1 = 0.7 there may betwo xn’s that it could have come from.
Example 6: Newton’s cooling equation
x = −α(x− xs),
is a continuous time system for any α, xs ∈ R. We may write it as a vector field
x = f(x),
where f(x) = −α(x − xs) with phase space X = R. Note that f is smooth (in fact smooth anddifferentiable) and so solutions are uniquely determined in both forward and backward time. Theevolution operator is given by Φt(x0) = (x0 − xs)e−αt + xs.
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Example 7: The planar pendulum equation
Mlθ + cθ +Mg sin θ = 0,
where M is the mass, l is the length of the pendulum, g gravitational constant and c the dampingcoefficient. By rescaling time t 7→
√l/gt, defining x = θ, y =
√l/gθ and letting D = c/m
√gl, yields
θ
θ
MgMgsinθ
M
l
Figure 6: Diagram of the planar pendulum
the equations of motion
x = y,
y = −Dy − sinx.
Since x = θ is the angular displacement from the vertical down position, this variable is periodicwith period 2π implying the phase space X is a cylinder. Alternatively, we can assume X = R2
and we will see the periodicity in x in the solutions. Given initial conditions on the initial angularposition θ(0) and initial angular velocity θ(0), solutions to the pendulum equations are uniquelydetermined for all time.
Example 8: The Lorenz system
x = σ(y − x), (12)
y = ρx− y − xz, (13)
z = −βz + xy, (14)
is a three-dimensional continuous dynamical system. Even though we cannot explicitly solve thesystem of ODEs, the solutions are uniquely determined in both forward and backward time by theinitial condition (x0, y0, z0) ∈ R3 in the phase space X = R3.
Example 9: Partial differential equations may also be considered as dynamical systems e.g., theheat equation
ut = ∇2u, u(x, 0) = g(x). (15)
The heat equation can be thought as a continuous dynamical system, T = R, where the phase spaceX is a function (Banach) space e.g., X = C2(R). In this case, the phase space also includes anyboundary conditions. One can also define an evolution operator which describes how the initialcondition u(x, 0) = g(x) evolves.
Everything that can happen in ODEs can also happen in PDEs but PDEs have far more degreesof freedom and so there is the possibility of far more complicated dynamics! For those that areinterested in dynamical systems methods applied to PDEs, see Nonlinear Patterns (MAT3022).
12
3 Key types of Solutions
From our early investigations into liquid cooling and population dynamics, it is clear there is aninitial transient part of the dynamics and then the dynamics “settles” down to, usually, three keytypes of solutions:
1. steady states,
2. periodic orbits and,
3. chaotic dynamics.
When dealing with nonlinear dynamical systems, we are less interested in the initial transient phaseand more in the long-time dynamics as these will govern how the system will evolve i.e., will youevolve to a steady state or a periodic orbit? One doesn’t often really care how you got there.
It is also extremely rare to be able to solve a nonlinear dynamical system analytically. Hence, we areless interested in precisely what the solution is doing compared to what the solution qualitatively(topologically) looks like. The three types of solutions define three qualitative types of possibledynamics (the list above is be no means exhaustive!). We then classify regions of parameter spacewhere the solutions look qualitatively similar i.e., steady states, periodic orbits, or chaotic dynamics.The first part of this course is concerned about how one can go from one type of qualitative behaviourto another.
3.1 Steady states
For ODEs, we define steady states (equilibria) to be where x = 0 i.e., the state does not change/evolveover time. An equilibrium x∗ can be found by solving the nonlinear problem
f(x∗) = 0, (16)
where f is the righthand-side of equation (10).
For maps, we define steady states (fixed point) to be where xn+1 = xn i.e., the state does notchange/evolve over time. A fixed point x∗ can be found by solve the nonlinear problem
x∗ = f(x∗), (17)
where f is the righthand-side of equation (11).
In both cases, one has to solve a nonlinear algebraic problem. While solving either (16) or (17) mayseem to be significantly easier than finding a solution for the general dynamics of (10) or (11), eventhis may be impossible to do!
Solution strategies for solving (16) or (17):
• Evolve the dynamical system and hope you converge to a steady state. Problem: there mightnot be a steady state or you might converge to something that isn’t a steady state. . .
• If the phase space is one-dimensional i.e., X = R, then one can look for steady states viagraphical inspection by:
1. plotting f(x) and looking for zeros of f in the case of ODEs, or
2. plotting f(x)− x and looking for zero crossings in the case of maps.
See example sheet 1 for examples of this.
• If the phase space is higher-dimensional, one can guess a steady state and use (a globalised)Newton’s method to solve (16) or (17). matlab’s fsolve routine in the optimisation toolboxwill do this. Problem: no guarantee of convergence to a steady state. . .
We will discuss steady states in more detail in the following sections, starting by understandingsteady states in ODEs.
13
3.2 Periodic orbits
Periodic orbits of dynamical systems are the simplest non-trivial evolving solutions.
For ODEs (10), we define a periodic orbit as follows: For a point x∗ ∈ X there exists τ ∈ R withτ > 0 such that Φτ (x∗) = x∗. The periodic orbit is defined as the closed curve
{Φt(x∗) | 0 ≤ t ≤ τ∗},
where τ∗ > 0 is the smallest number τ such that Φτ (x∗) = x∗. τ∗ is called the period of the periodicorbit.
For maps (11), we define a periodic orbit as follows: For a point x∗ ∈ X there exists n ∈ Z withn > 0 such that gn(x∗) = x∗, while gi(x∗) 6= x∗ for all 0 < i < n. The periodic orbit is defined asthe set of n points
{gi(x∗) | 0 ≤ i < n},and n is called the period of the periodic orbit.
Finding periodic orbits is even harder than finding steady states and so one usually has to resort tonumerical approximations.
3.3 Chaotic dynamics
Now this is where things start getting really interesting! Mathematicians can not even agree onwhat exactly chaos is. However, we are going to go for the following (vague) definition
Definition 2 (Chaos) Apparent stochastic (random) behaviour occurring in a deterministic sys-tem.
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n
xn
Logistic Map Lorenz Equations
Original initial conditionOriginal initial condition + small perturbation
x0
Figure 7: Examples of chaotic orbits in the logistic map and the Lorenz equations. In both systems, wehave started the systems off with an initial condition and let it run. Then we did the same computationbut this time with a very small perturbation to the initial condition (∼ 10−6). In both cases we see that theslightly perturbed initial condition evolves forward in time, initially staying close to the unperturbed evolution.Eventually though the orbits become very different! Matlab code: [Sensitive dependence on initial conditions]and [lorenz.m]
A dynamical system is chaotic if (a subset of the) orbits are confined to a bounded region, butbehave unpredictably (randomly). We have already seen one example of a chaotic system in theform of the logistic map xn+1 = 4xn(1 − xn). Another example is the Lorenz equations (14) with
14
σ = 10, ρ = 28, β = 8/3; see Figure 7. In this case, arbitrary orbits seem to accumulate on an objectcalled the butterfly attractor.
To demonstrate this unpredictability, we can start the chaotic dynamical system from an initialcondition, evolve it for some long time and remember the solution (orbit). Now, if the system waspredictable, we could re-start the dynamical system from the same initial condition plus a very smallperturbation and we would find that the new solution (orbit) would follow roughly the old orbit.But in chaotic systems what we observe is the following. We see that they behave similarly; inthe case of the logistic map the orbits stay bounded between [0, 1] and in the Lorenz equations theorbits trace the butterfly attractor. On the other hand, one would expect that two nearby initialconditions to trace out similar paths for all time, or at least for a very long time; e.g., the numberof turns in one of the butterfly wings to be followed by the same number of turns in the other wing.For chaotic systems this is not true as the orbits very quickly behave differently and there is no“memory” of the the fact that the initial conditions were once very close together. This propertyis known as sensitive dependence of initial conditions. In other words, the precision of yourinitial condition i.e., the number of decimal places, matters!
However, the logistic map (and the Lorenz equations) aren’t always chaotic. For example, if wechange to parameter r to r = 3.1 and evolve the logistic map xn+1 = rxn(1 − xn) then we find aperiodic orbit; see Figure 8.
0 5 10 15 20 25 30 35 40 45 500.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
n
xn
Figure 8: Evolution of the logistic map xn+1 = 3.1xn(1− xn) starting with x0 = 0.2. The orbit converges toa periodic orbit with iterates x0, x1, . . . , 0.765, 0.558, 0.765, 0.558, 0.765, 0.558, . . . , xn. Matlab code: [Periodicorbit of Logistic map]
So we have just changed a parameter from 4 to 3.1 (not a big change) and we have found verypredictable, regular behaviour. How to we know if a system is going to be regular or unpredictable?
Necessary conditions for chaos:
1. either
• a one- (or more) dimensional non-invertible iterated map (e.g. Logistic map), or
• a two- (or more) dimensional invertible iterated map (e.g. Henon map), or
• a three- (or more) dimensional system of (first order) differential equations (e.g. Lorenzequations).
2. nonlinearity!
So for example, the planar pendulum cannot be chaotic since it is only a system of two first-orderODEs even though it is nonlinear.
Some examples of differential equations that do have chaotic dynamics are:
15
M1
l1θ1
θ2
M2
l2
Example 10: Double Pendulum An extension of the planar pendulum is the double pendulum.Here we connect two planar pendulums together. Ignoring friction, we can write down the equationsof motion for this problem using Lagrangian dynamics. This yields the system of equations
(M1 +M2)l1θ1 +M2l2θ2 cos(θ1 − θ2) +M2l2θ22 sin(θ1 − θ2) + g(M1 +M2) sin θ1 = 0,
M2l2θ2 +M2l1θ1 cos(θ1 − θ2)−M2l1θ21 sin(θ1 − θ2) +M2g sin θ2 = 0
This system can be re-written as a first order ODE system of the form
θ1 = φ1,
φ1 = f1(θ1, θ2, θ2),
θ2 = φ2,
φ2 = f2(θ1, θ2, θ1)
Hence, we have enough dimensions and all the system is highly nonlinear. The dynamics of the doublependulum can be seen in this movie on the Chaos & Fractals website: [Double Pendulum].
Example 11: Chaotic Waterwheel The Chaotic Waterwheel shown in the movie on the Chaos& Fractals website: [Chaotic Waterwheel], is modelled by the Lorenz equation (14); see alsoChapter 9.1 of Nonlinear Dynamics and Chaos by Steven Strogatz. In the experiment, coloured wateris pumped into chambers (with holes in) in a cylinder that rotates. If the brake is set correctly, thewaterwheel will spin left and right chaotically.
16
4 Phase portraits and topological equivalence: flows
Before we can try to understand chaotic dynamics, we will start by understanding predictabledynamics. We just want to know roughly what the orbits of dynamical systems look like i.e., justdraw “sketches”. In particular, we are interested in what happens to the solutions as t → ∞. Forexample, the sketch of an orbit of a dynamical system going to an equilibrium should look verydifferent to the sketch of a dynamical system escaping to infinity. However, we want the sketches ofthe solutions going to the same equilibrium with different initial conditions to look the same. To dothis, we will use the idea of topological equivalence.
You will have come across the idea of phase portraits in your 2nd year Ordinary Differential equationscourse.
4.1 1D Phase portraits
Consider the linear one-dimensional ODE
x = αx (18)
with x ∈ R and a parameter α ∈ R. Solutions of (18) are simply x(t) = eαtx(0) and are uniquelydetermined by the choice of initial condition. In particular, we have the following qualitativelysimilar solutions
• If α > 0, x(t)→ ±∞ as t→∞ (x(t) escapes to +∞ if x(0) > 0 and −∞ if x(0) < 0)
• If α = 0, x(t) = x(0) for all t
• If α < 0, x(t)→ 0 as t→∞.
The case α = 0 is unusual and we will forget it here. x = 0 is an equilibrium solution of equation (18).
The phase portraits for (18) are shown in Figure 9 For all α < 0, the phase portraits all look the
x x
α<0 α>0
Figure 9: Phase portraits of (18). The equilibrium x = 0 is an attractor for α < 0 (i.e., all initial conditionsare converge to zero) and x = 0 is a repeller for α > 0 (i.e., all initial conditions except x(0) = 0 escape to±∞).
same and similarly for all α > 0.
In general for 1D ODEs of the formx = f(x),
one can quickly plot the phase portrait by plotting f(x) and everywhere f(x) > 0 the derivativex > 0 and hence the flow moves to the right, while if f(x) < 0 the derivative x < 0 and the flowmoves to the left.
Example 12: The phase portrait forx = sinx,
looks like Note that the phase portrait for x = sinx near x = 0 and x = 2π looks qualitativelysimilar to that for α > 0 in Figure 9. Similarly, near x = −π and x = π the phase portraits lookqualitatively similar to that for α < 0 in Figure 9.
We call these two systems topologically equivalent in these respective neighbourhoods.
17
-π
- 10 - 5 0 5 10
- 101
xf(x)<0
f(x)>0sinx
f(x)<0π
f(x)<0f(x)<0
f(x)>0 f(x)>0 f(x)>0f(x)=
Definition 3 (Topological Equivalence) Suppose we have two vector fields
x = f(x), (19)
y = g(y). (20)
Then (19) on domain U is topologically equivalent to (20) on a domain V , if we can find a continuousand invertible map (i.e., a homeomorphism) h : U → V that maps orbits of (19) to orbits of (20),respecting the direction of time.
So in their respective domains U and V , the vector fields have the same number of equilibria (onein our example) and identical phase portraits.
4.2 2D Phase portraits
We consider general autonomous two-dimensional dynamical systems of the form
x = f(x, y) y = g(x, y). (21)
Recall that an autonomous ordinary differential system is one where the right-hand-side does notdepend explicitly on time.
Example 13: Planar pendulum system
x = y =: f(x, y),
y = −Dy − sinx =: g(x, y),
where x = θ (angular position), D = c/m√gl .
Example 14: Competing Species model Population dynamics for two competing species x andy (that define fractions of an expected population size) for the same limited food resource such thatthey inhibit each others growth
x = x(1− x− αy) =: f(x, y), (22)
y = λy(1− y − βx) =: g(x, y). (23)
So the first species x would be considered healthy if x = 1, and similarly for the second species y = 1is a healthy population size. If either species is extinct, then the other species is governed by thelogistic equation (4).
Example 15: Harmonic oscillator Harmonic oscillator with a friction term. By defining y = xwe obtain the two-dimensional autonomous dynamical system
x = y,
y = −cy − ω2x.
18
M
xSpring
x + cx + ω2x = 0
The phase curves or phase trajectories of (21) are solutions of
dx
dy=f(x, y)
g(x, y). (24)
Through any point (x0, y0) there is a unique curve except at equilibria (xs, ys) where
f(xs, ys) = g(xs, ys) = 0.
Transform x 7→ x− xs, y 7→ y − ys then (0, 0) is an equilibrium of the transformed equation. Thus,without loss of generality, we now consider (24) at the equilibrium (0, 0); that is
f(x, y) = g(x, y) = 0 ⇒ x = 0, y = 0.
We expand both f and g is a Taylor series (using the fact that we have assumed enough smoothness,analytically to be precise), and retaining only the linear terms, we get
dx
dy=ax+ by
cx+ dy, A =
(a bc d
)=
(fx fygx gy
)
(0,0)
,
that defines the matrix A and the constants a, b, c, and d. The linear form is equivalent to the system
x = ax+ by, y = cx+ dy, (25)
whose solutions give parametric forms of the phase curves where t is the parametric parameter.
Let λ+ and λ− be the eigenvalues of A∣∣∣∣a− λ bc d− λ
∣∣∣∣ = 0 ⇒ λ± =1
2(a+ d± [(a+ d)2 − 4detA]1/2).
Solutions of (25) are then (xy
)= c1v+e
λ+t + c2v−eλ−t, (26)
where c1 and c2 are arbitrary constants and v+ and v− are the eigenvectors of A corresponding toλ+ and λ− respectively.
The form (26) is for distinct eigenvalues. If the eigenvalues are equal, the solutions are proportionalto (c1 + c2t)e
λt.
We can separate the phase portraits into three classes:
Attractors: Here both the eigenvalues have negative real part. If both eigenvalues are real andnegative then the phase portraits look qualitatively similar to
Eigenvalues Phase Portrait
C
x
y(a)
19
If the eigenvalues are complex with negative real part then the phase portrait looks like a spiral
C
x
y(b)Eigenvalues Phase Portrait
x>0y<0
.
.
To work out which way the spiral goes for a given system, just away from the origin calculate x andy. In the above phase portrait, I calculated the sign of x and y at the yellow dot and following thisinitial direction I can trace out the spiral.
Repellers: Here both the eigenvalues have postive real part. If both eigenvalues are real and postivethen the phase portraits look qualitatively similar to
C
x
y(c)Eigenvalues Phase Portrait
If the eigenvalues are complex with positive real part then the phase portrait looks like spirals
C
x
y(d)Eigenvalues Phase Portrait
Saddles: Here one eigenvalue is positive and the other is negative. The phase portrait qualitativelylooks like
C
x
y(e)Eigenvalues Phase Portrait
If there are no eigenvalues with zero real part, then we can say that the two phase portraits for theattractors are both qualitatively the same (similarly for the repellers)
Theorem 4.1 (Topological Equivalence for linear flows) Consider the two linear vector fields
x = Ax, (27)
x = Bx, (28)
20
where A and B are n×n matrices. If n0(A) = n0(B) = 0, that is the origin is a hyperbolic equilibriumfor both systems, then (27) and (28) are topologically equivalent if and only if n+(A) = n+(B) andconsequently, also n−(A) = n−(B).
This theorem is true for general linear systems of size n.
4.3 Stability and Eigenvalues
In general most vector fields will be high dimensional
x = f(x), x ∈ Rn, f : Rn → Rn, (29)
for some n ≥ 1 with f being nonlinear. Suppose we know an equilibrium xs that is
f(xs) = 0. (30)
We wish to find out if we start near the equilibrium state whether or not the solution x(t) convergesto xs as t→∞ i.e., xs is a stable steady state (see Figure ??(a) and (b)).
To do this, we consider the evolution of
x(t) = xs + x(t) (31)
where x(t) is assumed to be a small perturbation of the steady state xs. Now substituting (31) intothe ODE (29) yields
d[xs + x(t)]
dt= f(xs + x(t)),
dxsdt
+dx
dt= f(xs + x(t)),
dx
dt= f(xs + x(t)), (32)
since xs is a steady state with xs = 0.
Now we expand f(xs + x(t)) in a Taylor series remembering that x(t) is small
f(xs + x) = f(xs) + J(xs)x(t) + · · · (33)
where J(x) is the Jacobian n× n matrix of f defined as
J(xs) =
∂f1∂x1
· · · ∂f1∂xn
......
∂fn∂x1
· · · ∂fn∂xn
x=xs
, where x =
x1...xn
, f(x) =
f1...fn
.
Substituting the Taylor series for f (33) for the right hand side in (32) yields
dx
dt= f(xs) + J(xs)x(t) + · · · ,= J(xs)x(t) + · · · , since f(xs) = 0,
≈ J(xs)x(t), ignoring higher order terms in x(t).
This process of ignoring higher order (nonlinear) terms in x(t) is called linearisation. The stabilityof the equilibrium xs is governed by the linear ODE system
dx
dt= J(xs)x(t), (34)
whose solution is x(t) = x(0)eJ(xs)t (see first year Linear Algebra). The eigenvalues of the matrixJ(xs) are important for determining the stability properties of the equilibrium xs. We define thefollowing three numbers
21
• n0 = number of eigenvalues of J(xs) with zero real part,
• n+ = number of eigenvalues of J(xs) with positive real part,
• n− = number of eigenvalues of J(xs) with negative real part.
Definition 4 An equilibrium xs is called hyperbolic if n0 = 0. We call a hyperbolic equilibrium xsan attractor if n− = n and n+ = 0, a repeller if n+ = n and n− = 0, and a saddle point both ifn+ > 0 and n− > 0.
Attractors have the property x(t)→ 0 as t→∞ i.e.,
x(t) = xs + x(t)→ xs as t→∞
so all initial starting points near xs converge to xs and the equilibrium is said to be stable. On theother hand, repellers have x(t)→∞ as t→∞ then
x(t) = xs + x(t)→∞ as t→∞
and so the steady state xs is said to be unstable i.e., all initial starting points near xs diverge fromxs. Of course, if x(t) becomes large then this violates our Taylor series expansion requiring us totake into account higher order terms far from xs.
When does linearising about xs tell you the stability properties of xs? The Hartman & Grobmanntheorem tells you when the linearised system governs the stability of the equilibrium.
Theorem 4.2 (Hartman & Grobmann) If the vector field
x = f(x), (35)
has a hyperbolic equilibrium xs, then there exists a neighbourhood U of xs such that (35) on U istopologically equivalent to the linearised system
˙x = J(xs)x
on an (arbitrary) neighbourhood V of the origin.
Example 16: The system x = sinx near x = π is topologically equivalent to the system
x =
(d
dxsinx
)
x=π
x = (cosπ)x = −x.
Example 17: Harmonic oscillator
x = y,
y = −Dy − sinx.
Equilibria are given byy = 0 and sinx = 0 ≡ x = kπ, k ∈ Z
and the Jacobian matrix is
J
(xy
)=
(0 1
− cosx −D
).
Hence, for the equilibria (xs, ys) = (2kπ, 0) that matrix becomes
J
(2kπ
0
)=
(0 1−1 −D
),
with eigenvalues
λ± =−D
2± 1
2
√D2 − 4.
Now
22
• if D = 0, λ± = ±2i and so the equilibrium in not hyperbolic i.e., n0 = 2, n+ = 0, n− = 0.
• if 0 < D < 2, λ± are complex conjugate with negative real parts and the equilibrium ishyperbolic and an attractor i.e., n0 = 0, n+ = 0, n− = 2.
• if −2 < D < 0, λ± are complex conjugate with positive real parts and the equilibrium ishyperbolic and a repeller i.e., n0 = 0, n+ = 2, n− = 0.
For the equilibria (xs, ys) = (2kπ + π, 0) the matrix becomes
J
(2kπ + π
0
)=
(0 11 −D
),
with eigenvalues
λ± =−D
2± 1
2
√D2 + 4.
Hence, one eigenvalue is always positive and the other is negative ⇒ equilibria is a saddle point.
4.4 Stable and Unstable Manifolds
Linearisation tells us what happens to the dynamics locally (near) each hyperbolic equilibrium.However, we would like to know what happens globally to the dynamics i.e., how do we travel fromnear one equilibrium to another equilibrium? An important global object that will allow us to “paste”together local information near each equilibrium to obtain the entire phase portrait is the following.
Definition 5 Let xs be a saddle point of the vector field x = f(x). The set of all points “ending upat xs” under the flow of Φt of the vector field
W s(xs) = {x ∈ Rn | Φt(xs)→ xs as t→∞},is called the stable manifold of xs. Similarly, all points “coming from xs”
Wu(xs) = {x ∈ Rn | Φt(xs)→ xs as t→ −∞},is called the unstable manifold of xs.
The stable and unstable manifolds of saddle points in two-dimensional vector fields are one-dimensionalcurves. In general, the dimension of the unstable manifold is equal to n+ and the dimension of thestable manifold is n−. It is not easy to find stable and unstable manifolds; it is not possible, ingeneral, to compute them analytically. However, we do know the following theorem.
Theorem 4.3 Let xs be a saddle point of x = f(x). Then x0 is also a saddle point of the linearisedsystem
x = J(xs)(x− xs).Let Es(xs) and Eu(xs) denote the stable and unstable eigenspaces of the linearised system. In asmall enough neighbourhood U of xs, there exists a local piece of W s(xs), that is a smooth manifoldthat is a graph of some function h : Es(xs)→ Eu(xs). Furthermore, W s
loc(xs) is tangent to Es(xs)
at xs. The same is true for the local unstable manifold Wuloc(xs) which is defined in the same way.
Pictorially this looks like Figure 10.
Example 18: To illustrate global (un)stable manifolds and how to obtain the entire phase portraitlet us look again at the planar pendulum
x = y,
y = −Dy − sinx.
Remember that x = θ the angular displacement from the vertical down position and so x is periodicwith period 2π. The parameter D is proportional to the amount of friction in the system i.e., nofriction D = 0, friction D > 0.
We have already found the equilibria (xs, ys) = (kπ, 0) and stability
23
Es(xs)
Eu(xs)
Wuloc(xs)
W sloc(xs)
Figure 10: Diagram of stable and unstable manifolds near a saddle point.
• if (xs, ys) = (2kπ, 0):
– the eigenvalues are ±2i for D = 0 (no friction), and
– complex conjugate with negative real part for D small (small friction). The equilibria arespiral attractors.
• if (xs, ys) = (2kπ+π, 0), then one eigenvalue is always positive with the other always negativeand are saddle points with eigenvectors [1, 1]T and [−1, 1]T if D = 0.
The theory does not allow us to say anything about the equilibrium (xs, ys) = (2kπ, 0) for D = 0since the equilibria are non-hyperbolic. However, we know that a pendulum without friction willhave periodic motion about the vertical down position.
Hence, near the equilibria, the dynamics become
x
0 π 2π-π
x
0 π 2π-π
No Friction: D = 0
Friction: D >0
Figure 11: Phase portraits near the equilibria of the pendulum equation
If D = 0, then the periodic motion where the pendulum swings back and forth is separated fromthe periodic motion (the pendulum going over the top) by stable and unstable manifolds. If thependulum is (almost) exactly up (unstable situation) it will fall down and again end up (almost)exactly up. From this, the stable and unstable manifolds connect up as shown in Figure 12.
Note that there is a major change between the phase portraits for D = 0 (no friction) and D > 0(friction), but all friction phase portraits look qualitatively (topologically) the same for D > 0. Thismajor change in the phase portrait structure, is called a Bifurcation.
24
- 6 - 4 - 2 0 2 4 6- 3
- 2
- 1
0
1
2
3
x
y (a)
(b)
(c)
W s(xs)
Wu(xs)
(a) Pendulum goes over the top counter-clockwise
(b) Pendulum does not go over the top
(c) Pendulum goes over the top clockwise
Pendulum with no friction
- 6 - 4 - 2 0 2 4 6- 3
- 2
- 1
0
1
2
3 Pendulum with friction All typical initial conditions end up in the rest position
x
y
Figure 12: Global pendulum phase portrait with and without friction. Note, in the presence of friction,practically all initial conditions end up at the vertical down position. The exception is formed by the stablemanifolds, consisting of points that end up in the vertical up position. Maltab code: [pplane7.m]
5 Bifurcations: flows
Most models of dynamical systems contain parameters that we can either vary in an experimente.g., the carrying capacity K in the single species population model, or the parameters account fromsome uncertainty of the modelling. For instance, we do not know the exact value of friction in theplanar pendulum model, but we know that one exists. Hence the dynamical systems models are ofthe general form
x = f(x, λ), x ∈ Rn, and λ ∈ Rm.
The value of λ may lie in some interval, or region. Without knowing exactly what λ is, what canwe say about the dynamical system? We would like to separate out dynamical systems that behavequalitatively (topologically) the same e.g., in the competing species model
x = x(1− x− αy) =: f(x, y),
y = λy(1− y − βx) =: g(x, y).
for what parameters (α, β) causes the extinction of a species and for what parameters do the specieslive happily together?
The first thing to note is that the equilibria of the vector field
x = f(x, λ), x ∈ Rn, and λ ∈ Rm.
25
will typically change as λ is changed i.e., xs(λ) is a function of λ.
Example 19: In the single species population model
dN
dt= rN(1−N/K) = f(N, r,K),
the parameters here are r and K the carrying capacity of the environment. The equilibria are foundby solving f(Ns, r,K) = 0 for Ns where we find that Ns = 0 and Ns = K. The second of theseequilibria Ns = Ns(K) = K changes as K changes i.e., it is a function of K.
Example 20: Consider the predator-prey model for sharks and fish
F = p1F (F − 1)− FS − p2(1− e−p3F ), (36)
S = −S + p4FS, (37)
where F stands for the fraction of fish (between 0 and 1) and S is the fraction of sharks (also between0 and 1). The p1F (F − 1) term is the growth of fish similar to that for the single species populationmodel. The −FS term is sharks hunting fish and the p2(1− e−p3F ) term is people fishing. We havea fishing quota parameter p2 which we shall vary (we would not want the fish to become extinct)and the exponential term describes the fact that it is very hard to catch fish if the population isvery small. The growth term in the second equation p4FS describes that the number of sharks islimited by the number of fish. Finally, the −S term is the death of sharks.
We have the parameters:
• p1 - growth/birth rate of fish
• p2 - fish quota for people
• p3 - unknown parameter describing how hard it is to catch fish when the population is small
• p4 - how the sharks are limited by the number of fish
Let us find a equilibrium of this model i.e., we solve
0 = p1F (F − 1)− FS − p2(1− e−p3F ),
0 = −S + p4FS,
for F and S. For one equilibrium we get
F =1
p4, S =
p1p4 − p1 − p2p24 + p2e−p3/p4p24
p4.
So the equilibria depend on the parameters p1, p2, p3 and p4.
Definition 6 (Bifurcation) Consider the vector field
x = f(x, λ), x ∈ Rn, and λ ∈ Rm.
A bifurcation occurs at a parameter value λ = λb if for parameter values λ arbitrarily close, thephase portraits of the system are not topologically equivalent to those at λ = λb.
There are two main classes of bifurcations; local bifurcations and global bifurcations.
Definition 7 (Local Bifurcation) Consider the vector field
x = f(x, λ), x ∈ Rn, and λ ∈ Rm.
A local bifurcation occurs at (xb, λb) where (xb, λb) is an equilibrium i.e. f(xb, λb) = 0 and theJacobian J(xb, λb) has at least one eigenvalue with zero real part. The phase portrait of the systemis qualitatively (topologically) different for λ < λb and λ > λb.
26
There are many different types of bifurcations. We will being by classifying a few different typesof bifurcation that involve equilibria. These bifurcations are classified by the eigenvalues of theJacobian matrix associated with the parameter dependent equilibrium. At the bifurcation pointλ = λb, n0 6= 0 and the equilibrium is no longer hyperbolic. However, just before and after thebifurcation point, the equilibrium is hyperbolic and we can use the Hartman & Grobmann theorem.
Global bifurcations occur when larger invariant sets (e.g., periodic orbits) collide with equilibria.The change in the topology of the phase portraits is not confined to a local neighbourhood as is thecase of local bifurcations. We will discuss later examples of global bifurcations.
5.1 Saddle-Node Bifurcation
The saddle-node bifurcation in a system sees two equilibria on one side of the bifurcation but noequilibria on the other side as a single parameter is varied. At the bifurcation point, the two equilibriacome together and collide, annihilating each other. The bifurcation is sometimes also called a foldbifurcation, limit bifurcation, or turning point bifurcation.
Example 21: Consider the vector field
x = f(x, λ) = λ− x2,
where x, λ ∈ R. Solving f(x, λ) = 0, we find two equilibria x± = ±√λ. These two equilibria only
exist in R if λ > 0. The Jacobian matrix
J(x, λ) = −2x
and so we have
• at x+ = +√λ, the eigenvalue of the Jacobian matrix is −2
√λ < 0. So this equilibrium is an
attractor.
• at x− = −√λ, the eigenvalue of the Jacobian matrix is 2
√λ > 0. So this equilibrium is a
repeller.
There are several ways to visualise this bifurcation. One way would be to draw all the differenttopologically different phase portraits. However, we can combine all this pictures together to create
xxx
λ<0 λ=0 λ>0
−√λ
√λ0
one diagram, called a bifurcation diagram by plotting a graph with the parameter on the horizontalaxis and the phase space on the vertical axis. Taking vertical slices of the bifurcation diagram androtating the picture by 90 degrees yield the phase portraits shown above.
Similar to topologically equivalent phase portraits, we would like to identify qualitatively similarbifurcations.
Definition 8 (Normal forms) The (topological) normal form
y = g(y, λ)
of a vector field x = f(x, λ) is the simplest differential equation that captures the essential features ofa system near a bifurcation point i.e., in the neighbourhood of some point (xb, λb). The bifurcationdiagrams of both dynamical systems are topologically the same.
27
x
0 λ
x =√λ
x = −√λ
So the phase portraits of the normal form are topologically equivalent to those of the originaldynamical system near the bifurcation point.
Theorem 5.1 (Saddle-node bifurcation) Let
x = f(x, λ), (38)
with x, λ ∈ R. If the following conditions hold:
(B1) f(xb, λb) = 0 “Equilibrium at xb for λ = λb,”
(B2) J(xb, λb) = 0 “Zero eigenvalue at xb for λ = λb”,
(G1) d2
dx2 f(xb, λb) 6= 0 “Second order term of f does not vanish at equilibrium”,
(G2) ddλf(xb, λb) 6= 0 “Positive speed in λ”,
then (38) has the topological normal form
y = λ± y2,
in a neighbourhood of (xb, λb).
To make things more simple, the equilibrium (xb, λb) is shifted to the origin (x, λ) = (0, 0). Theconditions (B1) and (B2) are called bifurcation conditions and must be satisfied for the bifurcationto occur. (G1) and (G2) are called genericity conditions and they are usually satisfied (but must bechecked as well!).
Example 22: Consider the vector field
x = α− sinx = f(x, α),
where x, α ∈ R. Equilibria are found by solving f(x, α) = 0, i.e., α = sinx: graphically these canbe found by looking at intersections of the curves y = α and y = sinx. A possible candidate for a asaddle-node bifurcation is α = αb = 1 and x = xb = π/2. Check conditions:
(B1) f(xb, λb) = 1− sin(π/2) = 0, yep
(B2) J(xb, λb) = fx(αb, xb) = − cosxb = − cos(π/2) = 0, yep
(G1) d2
dx2 f(xb, λb) = sinxb = sin(π/2) = +1 6= 0, yep
(G2) ddλf(xb, λb) = +1 6= 0, yep.
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- 10 - 5 0 5 10
- 101
sinx
Candidates for saddle-node bifurcation x
Figure 13: Graphical picture of the equilibria α = sinx. The dashed lines correspond to α = +1 and α = −1.Varying α corresponds to moving these lines up or down. Equilibria correspond to where the dashed lineintersects the sin curve e.g, α = 0 curve is just the x-axis and the equilibria are shown as gold circles. Notethat at α = +1 and α = −1 we infinitely many collisions of pairs of equilibria. For |α| > 1 we have noequilibria i.e, at |α| = 1 we have a saddle-node bifurcation.
x
1α
π
2
Figure 14: Bifurcation diagram for x = α− sinx near the bifurcation point (αb, xb) = (1, π/2).
So at (αb, xb) = (1, π/2) we satisfy the conditions in the Saddle-node theorem and so we have asaddle-node bifurcation at this point. The bifurcation diagram looks like This picture looks thesame as the bifurcation diagram for y = λ + 1
2y2 near the bifurcation point (αb, xb) = (1, π/2).
Another way to see this is by first shifting x → x + π/2 to the origin and using the Taylor seriesexpansion for sin(x+ π/2) = cosx about x = 0
x = α−[1− 1
2x2 +
1
24x4 + · · ·
]= (α− 1) +
1
2x2 + · · ·
Now letting α− 1 = λ and x = y we see this is the same locally near the bifurcation point (xb, αb) =(π2 , 1) as y = λ+ 1
2y2.
5.2 Transcritical bifurcation
In a transcritical bifurcation, two equilibria “pass through each other” and exchange their stabilityproperties at the point where they collide. This bifurcation only occurs in systems where theequilibria exist for all parameter values.
Theorem 5.2 (Transcritical bifurcation) Let
x = f(x, λ), (39)
with x, λ ∈ R. If the following conditions hold:
(B1) f(xb, λb) = 0 for all λb ∈ R,
29
0 λ
x
Figure 15: Bifurcation diagram for the transcritical normal form y = λy − y2.
(B2) J(xb, λb) = 0 “Zero eigenvalue at xb for λ = λb”,
(G1) d2
dx2 f(xb, λb) 6= 0 “Second order term of f does not vanish at equilibrium”,
(G2) ddλfx(xb, λb) 6= 0 “Positive speed in λ”,
then (39) has the topological normal form
y = λy ± y2,in a neighbourhood of (xb, λb).
Example 23: In the single species population model
dN
dt= rN(1−N/K) = αN − βN2,
for α, β,N ∈ R, we see that this is topologically equivalent to the transcritical bifurcation normalform if β 6= 0 and there is a transcritical bifurcation at (Nb, α) = (0, 0).
5.3 Pitchfork bifurcation
If the dynamical system has a reflectional symmetry, then the usual bifurcation that will occur is apitchfork bifurcation with normal form
x = λx± x3.This system has reflectional symmetry about {x = 0} i.e., if you replace x with −x then you get thesame equation as before.
Theorem 5.3 (Pitchfork bifurcation) Let
x = f(x, λ), with f(−x, λ) = −f(x, λ) reflectional symmetry
with x, λ ∈ R. If the following conditions are met:
(B1) f(xb, λb) = 0,
(B2) J(xb, λb) = 0,
(G1) d3
dx3 f(xb, λb) 6= 0 “third order term of f does not vanish at equilibrium”,
(G2) ddλfx(xb, λb) 6= 0 “Positive speed in λ”,
then (38) has the topological normal form
y = λy ± y3,in a neighbourhood of (xb, λb).
When the cubic term is −x3, we call the pitchfork bifurcation supercritical; when the cubic term is+x3, the bifurcation is called subcritical.
30
x
0 λ
x =√λ
x = −√λ
x
0 λ
x =√−λ
x = −√−λ
x = λx − x3 x = λx + x3Supercritical pitchfork bifurcation Subcritical pitchfork bifurcation
5.4 Hopf bifurcation
The creation of small amplitude periodic orbits from an equilibrium is called a Hopf Bifurcation.This usually occurs when a complex conjugated pair of eigenvalues pass through the imaginary axis.The normal form for this bifurcation is
z = (λ+ iω)z + α|z|2z, “complex notation”. (40)
We may rewrite this equation in real variables x, y ∈ R by letting z = x+ iy
[xy
]=
[λ −ωω λ
] [xy
]+ α(x2 + y2)
[xy
]“real notation”. (41)
or in polar coordinates z = reiφ
r = r(λ+ αr2),
φ = ω. (42)
From (41), we that that if we linearise about (x, y) = (0, 0) then we have the system
[xy
]=
[λ −ωω λ
] [xy
]= A
[xy
].
The eigenvalues of A are λ± iω and so we require ω 6= 0 for the eigenvalues to be complex conjugate.They both pass through the imaginary axis at λ = 0.
It is easiest to analyse what happens in this system in the polar form (42) since the two equationsare uncoupled. Note that the phase space for (42) is the positive half cylinder r ≥ 0,−π ≤ φ ≤ π.Also, this system does not have any equilibria since φ is continuously varying with constant speedω i.e., φ(t) = ωt+ φ(0). Hence, everything flows to the right.
Lets first consider the case α = −1, ω > 0. The equation r = r(λ − r2) has three “equilbria”solutions, namely r = 0 and r = ±
√λ. From this the following phase portraits are
For λ > 0, we see the emergence of a periodic orbit. At the bifurcation point (xb, λb) the amplitudeof the periodic orbit is 0 and grows as
√λ. The frequency of the periodic orbit at the bifurcation
point is equal to ω, the absolute value of the imaginary part of the eigenvalue of the equilibriumpoint at bifurcation. The equilibrium exists for λ > 0 but is unstable while the periodic orbit isstable.
Note that if we ignore φ, then for the system in polar coordinates (42) the r = λr−r2 is the equationfor a pitchfork bifurcation with corresponding diagram. In the full parameter/ phase space R× R2
for the system of (41) the bifurcation diagram is a three-dimensional picture.
Theorem 5.4 (Hopf bifurcation) Let
x = f(x, λ), x ∈ R2, λ ∈ R. (43)
31
r
φ
λ<0
2π 2π 2π
λ=0 λ>0
0 0 0
√λ
``Slowly attracting"
In real (x,y) space:
λx
y r
0 λ
r =√λ
r = −√λ
(x,y,λ)-plane (r,λ)-plane
Suppose (43) has a equilibrium x(λ) for λ near λb with eigenvalues of the Jacobian matrix J(x(λ), λ)
η± = α(λ)± iβ(λ).
If the following conditions hold
(B1) f(xb, λb) = 0, (that is x(λb) = xb)
(B2) J(xb, λb) = ±iω, i.e., a pair of imaginary eigenvalues (α(λb) = 0, β(λb) = ω)
(G1) `1(xb, λb) 6= 0, where `1 is the first “Lyapunov quantity” and is computed from f (If you needit you should look it up in a book!)
(G2) ddλα(λb) 6= 0
then (43) has the topological normal form
z = (λ+ iω)z + `1(xn, λb)|z|2z
in a neighbourhood of (xb, λb).
The sign of the first Lyapunov quantity determines the stability properties of the periodic orbit thatappears in the bifurcation. We have the following overview
Example 24: Brusselator The Brusselator models an autocatalytic, oscillating chemical reaction.An autocatlytic reaction is one in which a chemical (c.f. species) acts to increase the rate of itsproducing reaction. In many autocatlytic systems complex dynamics are seen, including multiple
32
``Supercritical" ``Harmonic oscillator" ``Subcritical"!1(xb,λb) < 0 !1(xb,λb) = 0 !1(xb,λb) > 0
λ<0
λ=0
λ>0
steady-states and periodic orbits; see the Chemical Oscillations movie on the Chaos & Fractalswebsite [Chemical Oscillations]. The Brusselator system is given by the following system ofequations
x = A+ x2y − (B + 1)x,
y = Bx− x2y, (44)
where (x, y) ∈ R2 are two chemicals and A,B ∈ R are real parameters controlling the rates of thetwo chemicals. Let us fix A = 1. Equilibria correspond to when the amounts of the two chemicalsdoes not change. They are found by solving
1 + x2y − (B + 1)x = 0, Bx− x2y = 0, ⇒ x = 1, y = B.
We find the Jacobian matrix is
J(x, y) =
[2xy − (B + 1) x2
B − 2xy −x2],
evaluating the Jacobian matrix at the equilibrium (x, y) = (1, B) yields
J(A,B/A) =
[B − 1 1−B −1
],
with eigenvalues
η± =1
2B − 1± 1
2
√B(B − 4).
Hence, at B = 2, the eigenvalues are ±i and we have a candidate for a Hopf bifurcation. We havealready checked conditions (B1) and (B2). Condition (G2) is fulfilled since
d
dB(1
2B − 1) =
1
26= 0.
The hard condition to check is (G1) and this requires transforming the system into the normal form,equations (41). However, if we numerically integrate (44) for B = 2.1, we find a stable periodic orbitimplying that the first Lyapunov quantity is `1 < 0.
5.5 Global bifurcations
From a Hopf bifurcation small amplitude oscillations (periodic orbits) emerge from an equilibrium,but what happens to large amplitude periodic orbits? Well many things are possible; saddle-nodebifurcation of periodic orbits, periodic orbit collides with a saddle point (homoclinic bifurcation), ora period double bifurcation.
33
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
x
y periodic orbit
Figure 16: Convergence to the periodic orbit of the Brusselator. Matlab code: [Brusselator] and [brussela-tor.m]
5.5.1 Homoclinic bifurcation
A homoclinic bifurcation occurs when a periodic orbit collides with a saddle point. The bifurcationis easier to understand in an example.
Example 25: Homoclinic Bifurcation Consider the planar vector field
x = y,
y = λy + x− x2 + xy (45)
The homoclinic bifurcation in this system is found to numerically occur at λ = λh ≈ −0.8645. Forλ < λh, we have a stable periodic orbit. Note the stable and unstable manifolds of the saddle pointdo not coincide with each other.
As λ → λh from below, the periodic orbit collides with the saddle point, with the orbit spendingmore and more time at the saddle point. The period of the orbit scales like T ∝ ln(λ − λh) whichtends to ∞ as λ→ λh.
At λ = λh, a homoclinic orbit is formed. This is homoclinic orbits is formed by the stable andunstable manifolds of the saddle point coinciding.
For λ > λh, The bifurcation is described by the figure below.
Definition 9 (Homoclinic and Heteroclinic bifurcation) A homoclinic bifurcation is thecreation and destruction of a homoclinic orbit (i.e. an orbit that connects the unstable manifold ofa saddle point back to the stable manifold of the same saddle point), as a parameter is varied.
A heteroclinic bifurcation is the creation and destruction of a heteroclinic orbit (i.e., an orbitthat connects the unstable manifold of a saddle point to the stable manifold of another, differentsaddle point), as a parameter is varied.
Example 26: Heteroclinic bifurcation In the pendulum equations with no friction c = 0, wefind a heteroclinic orbit that connects the equilibrium (x, y) = (−π, 0) to the (x, y) = (π, 0) saddlepoint. Note, that there are in fact two heteroclinic orbits, the second orbit goes from (x, y) = (π, 0)back to (x, y) = (−π, 0).
Making friction c non-zero, destroys the heteroclinic orbit (i.e., a heteroclinic bifurcation).
For vector fields in 3+ dimensions, the mere crossing of the stable and unstable manifolds can implychaotic dynamics.
34
1
- 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2
- 2
- 1.5
- 1
- 0.5
0
0.5
1
1.5
2
x
y
- 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2
- 2
- 1.5
- 1
- 0.5
0
0.5
1
1.5
2
x
y
- 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2
- 2
- 1.5
- 1
- 0.5
0
0.5
1
1.5
2
x
y
- 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2
- 2
- 1.5
- 1
- 0.5
0
0.5
1
1.5
2
x
y 2
3 4
homoclinic orbit
periodic orbitperiodic orbit
Ws
Wu
Ws
Wu
Ws
Wu
Ws
Wu
Figure 17: Homoclinic bifurcation. Panel (1) Before the homoclinic bifurcation at λ = −0.92. We see aperiodic orbit. Note the stable and unstable manifolds of the saddle point do not coincide. Panel (2) Justbefore the homoclinic bifurcation at λ = −0.88, the periodic orbit starts to collide with saddle point. Panel(3) shows the homoclinic orbit at λ = λh ≈ −0.8645. Here the stable and unstable manifolds coincide andform a loop from the saddle point back to itself. Panel (4) shows the destruction of the periodic orbit forλ > λh. Matlab code: [pplane7.m]
A classic homoclinic bifurcation that creates chaotic dynamics is the destruction of a homoclinicorbit is a 3D saddle. In this 3D situation, the destruction of a homoclinic orbit can lead to thecreation of ∞-many periodic orbits!
The general system
x = −µ1 + f1(x, y, z, λ),
y = −µ2 + f2(x, y, z, λ),
z = µ3 + f3(x, y, z, λ),
where µi ∈ R. We assume µ1 > µ2. The bifurcation parameter for the system is λ. At λ = 0, wehave a homoclinic orbit to the saddle point.
We now define δ = µ2/µ3. If we change a parameter of the system λ, we destroy the homoclinicorbit. Depending on δ at the homoclinic bifurcation, chaos may be created.
1. If δ > 1, a stable symmetric periodic orbit exists for λ < 0 but two stable non-symmetric orbitsexist for λ > 0.
2. If δ < 1: no periodic orbits exist for λ < 0 and an unstable strange attractor (chaos) exists forλ > 0. This is called transient chaos and the system can have a normal attractor elsewhereto settle on.
Example 27: Homoclinic bifurcation in the Lorenz equations The chaotic dynamics in theLorenz equations (chaotic waterwheel) is created by a Homoclinic bifurcation. Recall the system isdefined by the following vector field
x = −σ(x− y),
y = ρx− y − xz,z = xy − βz.
35
- 6 - 4 - 2 0 2 4 6- 3
- 2
- 1
0
1
2
3
x
y
W s(xs)
Wu(xs)
Heteroclinic orbit
Heteroclinic orbit
Figure 18: Heteroclinic orbit. Matlab code: [pplane7.m]
Traditionally, we fix β = 83 , σ = 10 and vary ρ. The equilibria of the system are given by
x1 = (x1, y1, z1) = (0, 0, 0),
x2,3 = (x2,3, y2,3, z2,3) = (±√β(ρ− 1),±
√β(ρ− 1), ρ− 1).
The second two equilibria emerge at ρ = 1 in a supercritical pitchfork bifurcation off the origin. Forρ > 1, the origin x1 is a saddle point. The equilibria x2,3, are stable for 1 < ρ < ρH = 24.74. AtρH , the fixed points undergo a subcritical Hopf bifurcation.
If we now follow the emerging unstable periodic orbits from the Hopf bifurcation, and decrease ρ, wefind the periodic orbits get close to the origin equilibrium and at ρ ≈ 13.926 we have a homoclinicbifurcation.
unstable periodic orbit
homoclinic bifurcation
subcritical Hopf bifurcation
1 13.926 ρH=24.7424.06
transient chaos
strange attractor
ρ0 50 100 150 200
- 20
- 15
- 10
- 5
0
5
10
15
transient chaos
x2
T
x
Figure 19: Bifurcation diagram of the Lorenz system and an example of transient chaos. Matlab code:[Transient Chaos] and [lorenz.m]
At the homoclinic bifurcation ρ = 13.926, the eigenvalues of the origin equilibrium are
µ1 = −18.13, µ2 = −2.67 µ = 7.13,
and δ = 2.67/7.13 = 0.37 < 1. Hence, we have the emergence of transient chaos. Starting fromcertain initial conditions yields chaotic dynamics eventually decay to one of the stable equilibria x2,3.In this region, there is sensitive dependence on initial conditions and depending sensitively onthe initial conditions, one either ends at x2 or x3. It is important to note that even though theremay be stable equilibria, the dynamics of the system can still be very complicated!
36
This region of transient chaos lies in the parameter region 13.926 < ρ < 24.06, where the strange(chaotic) attractor stabilises at ρ = 24.06. Note there is co-existence of the stable chaotic attractorwith the stable fixed points for 24.06 < ρ < 24.74.
5.5.2 Period doubling route to chaos
To understand the period doubling bifurcation, we will look at a specific example.
Example 28: The Rossler system
x = −y − z,y = x+ ay,
z = b+ z(x− c), (46)
with (x, y, z) ∈ R3 and a, b, c ∈ R, is one of the simplest models that possesses period doublingbifurcations and chaotic dynamics. Notice that it is “almost” linear in that the only nonlinear term(xz) occurs in the third equation. Otto Rossler wrote down this system are being inspired by thesaltwater taffy machine that exhibits stretching and folding mixing of taffy; see the Taffy machinemovie on the Chaos & Fractals website [Taffy machine].
We set a = b = 0.1 and vary c (which we call the bifurcation parameter). We use Matlab tonumerically integrate these equations. Starting from a period 1, periodic orbit at c = 4, we increasec in steps. At c = 6, we see the period of the orbit has doubled (i.e., it goes around twice beforecoming back to the start). At c = 8.5 and c = 8.7, we see further period doubling of the orbit.Eventually, at c = 9 we see chaotic dynamics.
- 15 - 10 - 5 0 5 10 15- 15
- 10
- 5
0
5
10
15
- 20- 10
010
20
- 15- 10
- 50
510
150
5
10
15
- 15 - 10 - 5 0 5 10 15- 15
- 10
- 5
0
5
10
15
- 20- 10
010
20
- 15- 10
- 50
510
1502468
101214
- 10 - 5 0 5 10 15- 10
- 8
- 6
- 4
- 2
0
2
4
6
8
10
- 8 - 6 - 4 - 2 0 2 4 6 8- 8
- 6
- 4
- 2
0
2
4
6
- 10- 5
05
10
- 10- 5
05
100
0.5
1
1.5
2
2.5
3
c=4:period 1 c=6: period 2
c=8.5: period 4 c=8.7: period 8
x
y
xy
z
x
y
x
y
xy x
zy
xy
z
- 10- 5
05
1015
- 10- 5
05
1001234567
xy
z
- 15 - 10 - 5 0 5 10 15- 15
- 10
- 5
0
5
10
15
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010
20
- 15- 10
- 50
510
1502468
10121416
c=9: Chaos
y
z
xx
ySchematic of the chaotic attractor
Figure 20: Period doubling route to chaos in the Rossler system. The system undergoes an infinite sequenceof period doubling bifurcations as c is increased, eventually leading to chaos at c ≈ 9. The schematic pictureof the chaotic attractor is a Mobius strip. Matlab code: [Rossler simulations] and [rossler.m]
In order to analyse the period doubling bifurcation we will introduction a Poincare section. Wechoose a two-dimensional plane P , which is the Poincare section, defined by y = 0 say, with x, z > 0.This section must be transverse to the flow Φt. Suppose that we now take an initial condition inthe plane Σ given by
(x(0), y(0), z(0)) = x0.
37
Using this initial condition, we follow the trajectory of the differential equation under the flow Φt
until it again intersects Σ, when t = t1 say. Then we define the map
P : Σ→ Σ.
Iterating the map is equivalent to considering consecutive intersections of the orbit with Σ i.e.,
{Pn(x0) | n ∈ Z} = {Φt(x0) | t ∈ R} ∩ Σ
where Pn(x0) means applying P n times i.e., Pn(x0) = P (P (. . . P (x0))). Points “jump” on Σ underiteration of P . Instead of consideing the orbits of the vector field, we can now study the orbits of
0
z
0
0
xy
Φt
Poincaré section
x0
P (x0)
P (P (x0)) = P 2(x0)
Σ
Σ
Figure 21: A diagram showing the Poincare section and the flow intersecting at two points. Note the flowthrough the Poincare section is effectively one-dimensional.
points in Σ under the iteration of P . This is one of the most important techniques in dynamicalsystems theory.
Period Doubling Bifurcation Since solutions of the Rossler system are unique (see 2nd yearOrdinary Differential Equations course), trajectories cannot intersect itself. In 2D, we cannot havea period doubling bifurcation since the orbits would self-intersect.
The period-doubling in the Rossler system is due to a “stretch and fold” mechanism. Fold a to band a to a along the horizontal line. Then glue the ends together identifying a with a and b with b.Trajectories in the Rossler system are attracted very quickly to this object, after which they followit in the direction of the arrow. Since this attraction is very strong, we can make the approximationof the system’s behaviour by only considering this object and taking a Poincare section on it. Thisprocess results in the one-dimensional one-humped map.
To see this a bit better, we look in the chaotic region of parameter space, c = 9. We can explore this“stretching and folding” mechanism even further by using a cunning idea. . . lets look at successivelocal maxima of x(t) of the chaotic solution and plot these against each other. The result is analmost one-dimensional map that looks rather similar to the logistic map! This one-dimensionalmap is an excellent approximation of the dynamics on the Rossler attractor. Hence, we need tounderstand the dynamics of one-dimensional humped maps.
38
Σ
λ<0 λ=0 λ>0
x0P (x0)
ΣΣ
Σ Σ
1 2 3
45
Figure 22: Bifurcation diagram for a period doubling bifurcation. Matlab code: [Poincare map of Rosslersystem], [rossler-events-poincare.m] and [rossler.m]
0 10 20 30 40 50 60- 15
- 10
- 5
0
5
10
15
t
x(t)
xmax(N) xmax(N+1)
10.5 11 11.5 12 12.5 1313.9
14
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
x max(N+1)
xmax(N)
Figure 23: The “Next Maxiumum” return map. Approximate one-dimensional map of the chaotic Rossler sys-tem with a = b = 0.1, c = 9.Matlab code: [Rossler system 1D map], [rossler-events.-lorenz.m] and [rossler.m]
39
Example 29: An explicit Poincare map We consider the first-order order ODE
x+ x = cos(t).
This is an example of a non-autonomous dynamical system that we’ve not dealt with before. How-ever, we can re-write this ODE as a planar vector field by introducing an auxiliary ‘time’ coordinateτ where x = x(τ) and t = t(τ) and we have
xτ = cos(t)− x,tτ = 1,
with initial conditions x(0) = x0 and t(0) = 0. This system is equivalent since tτ = 1 ⇒ t = τ so tand τ are the same. This is now a two-dimensional vector field and we can have periodic orbits inthis system. The phase space of the system is on a cylinder (due to the cos(t) periodic forcing) thatlooks like
xt
�
intersections of�
We define the Poincare section Σ to be
Σ = {(x, t) : t = 0 mod 2π} .
Hence, the time of flight between successive intersections of the Σ is T = 2π.
The general solution of the ODE can be found via an integrating factor:
(etx)t) = et cos(t),
⇒ x(t) = e−t∫ t
0
es cos(s)ds+ cet.
Now from integration-by-parts we have
∫ t
0
es cos(s)ds = [es sin(s)]t0 −
∫ t
0
es sin(s)ds,
and also from integration-by-parts, we have
∫ t
0
es cos(s)ds = [es cos(s)]t0 +
∫ t
0
es sin(s)ds.
Hence, we have
2
∫ t
0
es cos(s)ds = et(sin(t) + cos(t))− 1,
and the general solution (using the initial conditions x(0) = x0) is given by
x(t) =1
2(cos(t)− sin(t)) +
e−t
2+ x0e
−t.
40
Therefore, the Poincare map P : Σ 7→ Σ is given by
P (x0) = x(2π) =1 + e−2π
2+ x0e
−2π
(Note, we can replace x0 by another variable, say, y). The cobweb of the map is plotted below wherewe see that there is a unique, stable fixed point of the map that corresponds to a stable periodicorbit of the ODE
yy = x
x
P(x)
To prove that P as a globally stable fixed point, we use the contraction mapping theorem (seeIntroduction to Function Spaces). The theorem states that if P : R 7→ R and
|P (x)− P (y)| ≤ k|x− y|, where k < 1,
then P has only one fixed point and the iteration xn+1 = P (xn) converges to it i.e., the fixed pointis globally stable. So we need to compute
|P (x)− P (y)| = |e−2π(x− y)| = e−2π|x− y|,
and we note that e−2π < 1. Hence, we can use the contraction mapping theorem to prove theexistence of a unique, globally stable fixed point that corresponds to a unique, stable periodic orbitof the ODE system.
41
6 One-dimensional maps
From the previous section, we saw that the dynamics of the Rossler system are well approximatedby a one-dimensional humped (parabolic) map, a bit like the logistic map xn+1 = rx(1− x).
Let us start by trying to understand linear maps.
Example 30: Exponential decay Consider the map
xn+1 =1
2xn = g(xn)
with phase space X = R and evolution operator given by Φt = Φn(x) = gn(x). Lets take x0 = 2,then we find the following orbit
x0 = 2, x1 = 1, x2 =1
2, x3 =
1
4, x4 =
1
8, . . . , xn =
(1
2
)n2.
We see that the iterates are getting smaller, in fact as n→∞ xn → 0. This is true regardless of theinitial condition x0 taken. The point x = 0 is a fixed point of the map. Fixed points are the mapequivalent of equilibria in vector fields.
Definition 10 (Fixed points) Fixed points are found by solving
g(x) = x,
i.e., if we apply the flow operator Φt to x then we stay where we are.
Solving this equation for g(x) = 12x = x we find that x = 0 is a fixed point of the linear map. Similar
to stable equilibria in vector fields, we call x = 0 an attractor.
We can visualise this by drawing a phase portrait similar to that for one-dimensional odes.
1
4
1
2
0 1 2xn
xn+1 = xn
xn+1 =1
2xn
x0
1
xn+12
x1x2
(b)
x0x1x2x3
1
4
1
20 1 2
x
(a)
Figure 24: (a) Phase portrait for the linear map xn+1 = 12xn, x0 = 2 converging to the fixed point x = 0.
(b) Cobweb diagram showing the iterates. Starting at x0, we follow the gold arrow till we hit the blue curve(xn+1 = 1
2xn) to find the next iterate x1 reading its value on the vertical axis by following the black arrow.
To find the next iterate x2, we then follow the second gold arrow and read of the value on the horizontalaxis. Iterating the map follows the gold and black arrows. Fixed points of maps (not just linear maps!) maybe graphically found at points where the curves xn+1 = f(xn) (blue curve) intersect the xn+1 = xn (gray)curve.
Note that the linear mapxn+1 = λxn,
with −1 < λ < 1 always has the fixed point x = 0 which is an attractor. The corresponding phaseportraits are qualitatively similar to Figure 24.
42
Example 31: Exponential growth Consider the map
xn+1 = 2xn = g(xn)
with phase space X = R and evolution operator given by Φt = Φn(x) = gn(x). Lets take x0 = 14 ,
then we find the following orbit
x0 =1
4, x1 =
1
2, x2 = 1, x3 = 2, x4 = 4, . . . , xn = 2n
1
4.
We see that the iterates are getting larger, in fact as n → ∞ xn → ∞. This is true regardless ofthe initial condition except x0 = 0. Again x = 0 is a fixed point of the map, but in this case it is arepeller.
We can visualise this by drawing a phase portrait similar to that for one-dimensional odes.
1
4
1
2
0 1 2xn
xn+1 = xn
1
xn+12
x2 x3x0
x1
(b)xn+1 = 2xn
1
4
1
20 1 2
x0 x1 x2 x3
x
(a)
Figure 25: (a) Phase portrait for the linear map xn+1 = 2xn, x0 = 14
.We see that all initial conditions closeto x = 0 escape to infinity. (b) Cobweb diagram showing the iterates. Starting at x0, we follow the goldarrow till we hit the blue curve (xn+1 = 1
2xn) to find the next iterate x1 reading its value on the vertical axis
by following the black arrow. To find the next iterate x2, we then follow the second gold arrow and read ofthe value on the horizontal axis. Iterating the map follows the gold and black arrows.
Note that the linear mapxn+1 = λxn,
with |λ| > 1 i.e., λ < −1 or λ > 1, always has the fixed point x = 0 which is a repeller. Thecorresponding phase portraits are qualitatively similar to Figure 25.
6.1 Cobwebs
A useful tool in analysing 1D maps xn+1 = f(xn) are cobwebs. The iteration xn+1 = f(xn) can wewritten in two stages as
yn = f(xn),
xn+1 = yn.
By drawing the two curves y = f(x) and y = x, the iteration can be represented graphically asfollows. Given an initial condition x0, draw a vertical line until it intersects the graph y = f(x);that height is the output y0. Now draw a horizontal line till it intersects the diagonal line y = x; thispoint is now x1 = y0. To compute x2, now move vertically to the curve again. Repeat the process ntimes to generate the first n points. Cobwebs are useful since they allow us to see global behaviourat a glance allowing us us to piece together the linearised dynamics similar to phase portraits forvector fields.
The figures below show the iterate and cobweb for the Logistic map xn+1 = 2xn(1 − xn) withx0 = 0.1. These figures are made with the Matlab code [cobweb.m]
43
x0 x1 x2
x1
x2
yn = f(xn)
xn+1 = yn
5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n
xn
x
y = f(x)
y = x
6.2 Topological conjugacy and linearisation
Similar to vector fields, we can define topological equivalence for maps. For maps however, a weakerproperty that is useful is topological conjugacy.
Definition 11 (Topological Conjugacy: maps) Suppose we have two maps
xn+1 = f(xn), (47)
yn+1 = g(yn). (48)
Then (47) on domain U is topologically conjugate to (48) on a domain V , if we can find a continuousand invertible map (i.e., a homeomorphism) h : U → V that maps orbits of (47) to orbits of (48)i.e.,
h(f(x)) = g(h(x)).
There is a slightly weaker property of topological conjugacy called topological semi-conjugacy inwhich the map h is only continuous and onto.
Topological Equivalence is when the maps are topologically conjugate and the direction of time isrespected between the orbits from the maps f and g.
Suppose we now have a general nonlinear map
xn+1 = f(xn), x ∈ R, f : R→ R,
and we can find a fixed point f(xf ) = xf . We wish to now if this fixed point is attracting or repelling.To do this we will consider the evolution of a small perturbation of the fixed point
xn = xf + xn,
44
where |xn| � 1 (i.e., very much less than 1). Now substituting this into our map yields
xf + xn+1 = xn,
= f(xn),
= f(xf + xn),
≈ f(xf ) + fx(xf )xn, by Taylor series expansion of f(xf + xn),
= xf + fx(xf )xn, since xf = g(xf ) is a fixed point.
This leads to the simple linear map
xn+1 = fx(xf )xn = µxn,
which is a good approximation to f near to the fixed point xf provided that xn is small. µ is calleda Floquet multiplier and is similar to an eigenvalue of the Jacobian for ODEs.
Now we know that if
• −1 < µ < 1, then xn → 0 as n→∞. Hence xn → xf as n→∞ and xf is called an attractor.
• |µ| > 1, then xn → ∞ as n → ∞. Hence xn becomes large and diverges from xf . We call xfa repeller.
When is linearisation of maps ok? Similar to differential equations we have the following version ofthe Hartman & Grobmann theorem to tell us.
Theorem 6.1 (Hartman & Grobmann : maps) If the map
xn+1 = f(xn), (49)
has a hyperbolic fixed point xf , that is |fx(xf )| 6= 1, then there exists a neighbourhood U of xf , suchthat (49) on U is topologically equivalent to the linearised system
xn+1 = fx(xf )xn,
on an (arbitrary) neighbourhood V of the origin.
At µ = 1 or µ = −1, we have a bifurcation.
Definition 12 (Local bifurcation: maps) Consider the map
xn+1 = f(xn, λ), x, λ ∈ R.
A local bifurcation occurs at (xb, λb), where (xb, λb) is a fixed point i.e., f(xb, λb) = 0, then
|fx(xb, λb)| = 1.
The phase portraits of the map are qualitatively (topologically) different for λ < λb and λ > λb.
As with vector fields, we also have a saddle-node bifurcation (the creation of two new equilibria)and a transcritical bifurcation (the old equilibrium remains and a new equilibrium is created)
Theorem 6.2 (Saddle-node bifurcation: maps) Let
xn+1 = f(xn, λ), (50)
with x, λ ∈ R. If the following conditions hold:
(B1) f(xb, λb) = xb “xb is a fixed point for λ = λb”,
(B2) fx(xb, λb) = 1, “Unit eigenvalue at xb for λ = λb”,
45
(G1) fxx(xb, λb) 6= 0 “Second order term of f does not vanish at fixed point”,
(G2) fλ(xb, λb) 6= 0 “Positive speed in λ”,
then (50) has the topological normal form
yn+1 = λ+ yn ± y2n,
in a neighbourhood of (xb, λb).
Theorem 6.3 (Transcritical bifurcation: maps) Let
xn+1 = f(xn, λ), (51)
with x, λ ∈ R. If the following conditions hold:
(B1) f(xb, λb) = xb for all λb ∈ R, “xb is a fixed point for all λ”,
(B2) fx(xb, λb) = 1, “Unit eigenvalue at xb for λ = λb”,
(G1) fxx(xb, λb) 6= 0 “Second order term of f does not vanish at fixed point”,
(G2) fxλ(xb, λb) 6= 0 “Positive speed in λ”,
then (51) has the topological normal form
yn+1 = (1 + λ)yn ± y2n,
in a neighbourhood of (xb, λb).
Example 32: Saddle-node bifurcation Consider the map
xn+1 = f(xn, λ) = λexn , xn ∈ R, λ ∈ R.
Fixed points of this map can be found graphically by plotting the graphs y = x and y = λex, andlooking for intersections of the graphs. For λ > 1
e , the graphs do not intersect. At λ = 1e , there is
- 1 - 0.5 0 0.5 1 1.5 2 2.5 3- 1
0
1
2
3
4
5
λ
y = x
fixed points
y = λex
x
1
1
e
λ
one intersection (a fixed point) and for λ < 1e there are two intersections (two fixed points). This
indicates that there is a saddle-node bifurcation at (x, λ) = (1, 1e ) where two fixed points are eithercreated (as λ is decreased) or destroyed (as λ is increased).
We can check the bifurcation conditions at (x, λ) = (1, 1e )
(B1) f(1, 1e ) = 1,
(B2) fx(1, 1e ) = 1,
(G1) fxx(1, 1e ) = 1 6= 0,
46
(G2) fλ(1, 1e ) = e 6= 0.
Example 33: Logistic map: transcritical bifurcation Consider the logistic map
xn+1 = rxn(1− xn) =: f(xn), xn ∈ [0, 1], r ∈ [0, 4].
We find two fixed points
f(xf ) = xf , ⇒ xf = 0 or 1− 1
r,
only the first of which depends on the parameter r. Hence we satisfy (B1). Note that the secondfixed point only exists for r > 1 and at r = 1 both fixed points are identical indicating a transcriticalbifurcation.
We find the stability of these fixed points by computing fx(x) = r(1 − 2x) and evaluating at thefixed points we find
fx(0) = r ⇒ xf = 0 is stable provided 0 ≤ r < 1,
fx(1− 1
r) = 2− r ⇒ xf = 1 is stable provided 1 < r < 3.
So at r = +1, we have a bifurcation of both points. For the fixed point x = 0, we have fx(0, 1) = +1satisfying (B2). The genericity conditions are also satisfied
(G1) fxx(0, 1) = −2 6= 0,
(G2) fxr(0, 1) = 1 6= 0.
Hence we have a fixed point bifurcation. The new emerging fixed point is in fact just x = 1− 1r .
0 10
1
xn+1
xn
r < 1
r = 1
r > 1
x
0 r1
x = 1 − 1
r
(a) (b)
Figure 26: (a) Cobweb diagram showing the emergence of a new fixed point for r > 1. (b) Bifurcationdiagram of the logistic map.
Top tip: You can quickly tell the difference between a saddle-node bifurcation and a transcriticalbifurcation (without checking the bifurcation conditions) by drawing the graphs y = x and y =f(x, λ) and looking for intersections (corresponding to fixed points). If you see two fixed pointsbeing created/destroyed as λ is varied, it is a saddle-node bifurcation. If you only see one new fixedpoint being created then it is a transcritical bifurcation.
47
6.3 Period doubling bifurcation
Definition 13 (Periodic orbits: maps) We define a periodic orbit in a map
xn+1 = f(xn)
to be the sequence of iteratesx0, x1, · · · , xn−1, n <∞,
where each iterate xi is not equal to any other iterate in the sequence and fn(x0) = x0. We say x0is a period n point i.e., xn = x0
Note that we may consider periodic orbits to be fixed points of the map xn+1 = fn(xn).
How does one go from a fixed point to a periodic orbit? Via a period doubling bifurcation (sometimescalled a flip bifurcation).
Theorem 6.4 (Period doubling bifurcation) Let
xn+1 = f(xn, λ), (52)
with x, λ ∈ R. If the following conditions hold:
(B1) f(xb, λb) = xb
(B2) fx(xb, λb) = −1,
(G1) −2fxxx(xb, λb)− 3(fxx(xb, λb))2 6= 0
(G2) c = fλfxx + 2fxλ(xb, λb) 6= 0,
then (52) has the topological normal form
yn+1 = −(1 + λ)yn ± y3n, (53)
in a neighbourhood of (xb, λb). If c > 0 then the bifurcation is said to be subcritical with the normalform
yn+1 = −(1 + λ)yn − y3n,while the bifurcation is supercritical if c < 0 with the normal form
yn+1 = −(1 + λ)yn + y3n.
Example 34: Logistic map Lets examine the logistic map again
xn+1 = f(xn, r) = rxn(1− xn), λ ∈ R
has the fixed point x = 1− 1r that is stable for 1 < r < 3. At r = rb = 3, the fixed point is x = 2
3 andwe find fx(xb, rb) = fx(2/3, 3) = −1. Hence we satisfy conditions (B1) and (B2). For the genericityconditions we find
(G1) −2fxxx(xb, rb)− 3(fxx(xb, rb))2 = −18 6= 0
(G2) fr(xb, rb)fxx(xb, rb) + 2fxr(xb, rb) = − 53 < 0 ⇒ hence the bifurcation is supercritical.
The new emerging periodic orbit is found by finding fixed points of the map f2(x, r) = f(f(x, r), r)i.e., solving
f(f(x, r), r) = x,
r[rx(1− x)](1− [rx(1− x)]) = x,
⇒ r3x4 − 2r3x3 + (r3 + r2)x2 + (1− r2)x = 0.
48
Now in general, it is very hard to solve quartic polynomials. However, there is a trick that willhelp us. . . any fixed point of f(x, r) is also a fixed point of f2(x, r) since f(f(x, r), r) = f(x, r) = x.Hence the fixed points of the logistic map are roots of the above quartic polynomial and we canfactorise to find
r3x4 − 2r3x3 + (r3 + r2)x2 + (1− r2)x = P (x, r)[f(x, r)− x]
where P (x, r) = r2x2 − (r2 + r)x+ (r + 1). Finding roots of P (x, r) yields
xp+,p− =r − 1
2r± 1
2r
√(r + 1)(r − 3)
The two solutions are the new periodic orbit. It can be shown that this periodic orbit is stable forr ∈ (3, 1 +
√6) by considering the map xn+1 = f2(x, r) and linearising this map about the above
roots of P (x, r).
In fact the logistic map has an infinite sequence of period doubling bifurcations eventually resultingin chaotic dynamics. This is one of the most common ways in which both ODEs (for instance theRossler system (46) and Lorenz system (14)) and maps go from order to chaos; known as the perioddoubling route to chaos.
2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.00.0
0.2
0.4
0.6
0.8
1.0
x
x = 1 − 1
r
period doubling bifurcations
xp+
xp−
chaos
periodic window
rFigure 27: Bifurcation diagram of the logistic map showing the infinite sequence of period doubling bifurcationsaccumulating at r = r∞ = 3.5699456 . . . corresponding to the onset of chaotic dynamics. Matlab code:[Logistic map bifurcation diagram]
The values of the period doubling bifurcations for a 2n-periodic orbit are
r1 = 3 period 2-orbit is born,
r2 = 1 +√
6 = 3.449 . . . 4,r3 = 3.54409 . . . 8,r4 = 3.5644 . . . 16,r5 = 3.568759 . . . 32,
......
r∞ = 3.569946 . . . ∞
49
We see that successive bifurcations occur closer and closer together, converging to a limiting valuer∞. If we calculate the distance between successive period doubling bifurcations we find it distanceshrinks by a constant factor (called Feigenbaum’s constant)
δ = limn→∞
rn − rn−1rn+1 − rn
= 4.6692 . . .
Amazingly, we see this factor and period doubling route in many other one-hump maps e.g., xn+1 =r sin(πxn).
Beyond r∞ chaotic dynamics are observed but not for all values r > r∞. For example, at approxi-mately r = 3.83, we see a periodic window, containing a stable period-3 orbit.
6.4 Periodic windows and Intermittency
Lets us examine in more detail, the periodic window containing a stable period-3 orbit at approxi-mately r = 3.83. This is the largest of the periodic windows, and they are all created in much thesame way. Hence, we will concentrate on the biggest periodic window. Looking at the bifurcation
3.82 3.83 3.84 3.85 3.860.1
0.5
1
r
x
diagram, Figure 27, we see that as r is increased (in the periodic window) there is another perioddoubling “cascade” to chaos. This period doubling sequence is qualitatively similar to the one wehave seen before. This qualitatively similarity in the bifurcation is known as self-similarity andwe will explore this further later.
Also, as r is decreased, at r ≈ 3.828we see an immediate change to chaos rather than a perioddoubling cascade.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
x
fixed pointsperiod-3 points
y = f3(x)
r = 3.8 r = 3.84
So how is a stable period-3 orbit created? A period-3 orbit satisfies the polynomial
x = f3(x) = f(f(f(x))),
50
where f(x) = rx(1 − x). Trying to analytically solve this polynomial problem is a bit tricky sincethe polynomial is of degree 8. Instead, we will use the graphical method of looking for intersectionsof the graphs y = x and y = f3(x).
We see at r = 3.84, we see 8 intersections of the graphs y = x and y = f3(x): two of these arefixed points, the other six period-3 points (3 stable, 3 unstable). Decreasing r to r = 3.8, we see the6 period-3 points have disappeared and there must have been a saddle-node bifurcation. It can beshown the saddle-node bifurcation occurs at r = 1 +
√8 = 3.8284. Just below this value for r (say
at r = 3.8282), the logistic map exhibits intermittency; chaotic dynamics looks a bit like a stableperiod-3 orbit for a short time. However, the period-3 orbit doesn’t exist! We are seeing a ghost ofthe period-3 orbit. . .
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n
xn
nearly period-3 orbitchaos
Figure 28: Chaotic orbit of the Logistic map for r = 3.8282 showing intermittency. Matlab code: [Intermit-tency]
Graphically, we can understand this intermittency by looking at the graphs y = x and y = f3(x)at r = 3.8282 and plotting a cobweb trajectory. We see there are three narrow “channels” betweenthe y = f3(x) graph and y = x. These channels get bigger as r is further decreased away fromthe saddle-node bifurcation. Zooming-in near one of these channels, we see an orbit may take manyiterations to pass through the small channel. In this channel, xn ≈ f3(xn) and so the orbit resemblesthe period-3 orbit. Eventually, the trajectory passes through the channel and becomes chaotic until
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.49 0.495 0.5 0.505 0.51 0.515 0.52 0.525 0.530.48
0.49
0.5
0.51
0.52
0.53
0.54
x
y
x
y = f3(x)
Figure 29: Cobweb diagram of the cubic composition of the Logistic map i.e. xn+1 = f3(xn, r) at r = 3.8282.Matlab code: [Cobweb Intermittency]
it comes back to one of the small channels.
This intermittent chaos is observed in many systems where the transition from periodic to chaoticbehaviour occurs due to a saddle-node bifurcation of periodic orbits. Intermittency is observed inexperimental systems where nearly periodic motion is interspersed with irregular busts of aperiodic“chaotic” motion. As the experimental control parameter is moved away from the saddle-nodebifurcation, the bust become more frequent until the system becomes fully chaotic. This is knownas the intermittency route to chaos.
51
This intermittent behaviour is typical of systems near a saddle-node bifurcation. In order to analysethis a bit, we first will look at the normal form for a saddle-node bifurcation in 1D
x = r − x2.
Taking r < 0 we have no equilibria. We can solve this ODE with x(0) = x0 to find
x(t) = tan
(t√−r + arctan
[x0√−r
])√−r.
Hence, we find
t√−r = arctan
(x0√−r
)− arctan
(x(t)√−r
).
Now, we can use this to find the time it takes x(t) to go from x = 1 to x = −1 i.e,
t√−r = 2arctan
(1√−r
),
and as r → 0 from below, we find
t ≈ 2π√−r .
Hence, as r gets smaller and smaller the length of time spent in the interval x ∈ [−1, 1] grows.
x
0 λ
x =√λ
x = −√λ
{t ∼ 1√−r
So what does this calculation have to do with the intermittency seen in a map? We can approximatethe continuous derivative with a discrete approximation
x ≈ xn+1 − xn∆t
,
Hence, substituting this into x = r − x2, we find
xn+1 = xn + ∆t(r − x2n),
and we would expect the dynamics of this map to closely match those of the continuous ODEprovided ∆t is small. If we employ the rescaling xn → xn/∆t, we find this map reduces to thenormal form for a saddle-node in maps
xn+1 = r + xn − x2n.
The bifurcation diagram for this map is the same as for the ODE and we would then expect thetime spent in the interval x ∈ [−1, 1] to also approximately scale like t ∼ 1/
√−r.
52
6.5 Period 3 Implies Chaos
The existence of a period-3 orbit in a one-dimensional continuous map allows us to say far moreabout the existence of other periodic-orbits!
Theorem 6.5 Consider the one-dimensional map
xn+1 = f(xn).
If f is a continuous function and the map has a period-3 orbit, then the map also has periodic orbitsof all other periods.
This theorem says nothing about the stability of all the other periodic orbits (hence, we don’tnecessarily see them in the period-3 window). It can also be shown that the existence of a period-3orbit implies the existence of infinitely-many orbits of the map that are not periodic.
Before proving this theorem, we shall first prove two other lemmata.
Lemma 1 Let I = [a, b] denote an interval and f be a continuous map. If f(I) ⊃ I then f has afixed point in I.
Proof. Let f(I) = [c, d] denote the interval that f maps I to. Since f(I) ⊃ I, we have
c ≤ a < b ≤ d.
Furthermore, there exists constants α ≥ a and β ≥ b such that
f(α) = c, f(β) = d.
To show that there exist a solution of the fixed point problem f(x) = x, we consider the functiong(x) = f(x) − x and we wish to show that there exists an x∗ such that g(x∗) = 0. To do this wenote,
g(α) = f(α)− α,= c− α,≤ c− a,≤ 0.
and
g(β) = f(β)− β,= d− β,≥ d− b,≥ 0.
Now, by the Intermediate Value Theorem (see MAT2041 Pure 1 (Real Analysis 2)), there existsan x∗ between α and β such that g(x∗) = f(x∗)− x∗ = 0 and this is a fixed point of f .
Corollary 1 If fn(I) ⊃ I, then fn has a fixed point in I and hence f has a period-n point in I.
Lemma 2 Let f be a continuous map. If I and J are two closed intervals such that g(I) ⊃ J , thenthere exists a closed subinterval I ′ ⊂ I such that f(I ′) = J .
Proof. Let I = [a, b] and J = [c, d]. Now c ∈ J ⊂ f(I) and so there exists a′ ∈ I such thatf(a′) = c. Similarly, d ∈ f(I) and so there exists b′ ∈ I such that f(b′) = d. There may in factbe several values a′, b′ satisfying this condition. If b′ ≥ a′, then we define I ′ = [a′, b′], otherwise wedefine I ′ = [b′, a′]. Now we may choose a′ and b′ such that there does not exist a γ ∈ (a′, b′) suchthat f(γ) = c or d otherwise we could re-define a′ = γ if f(γ) = c or b′ = γ if f(γ) = d. Hence,f(I ′) = J .
53
We are now in a position to prove theorem 6.5.
Proof of Theorem 6.5
We start by considering the period-3 orbit of the map f and we denote x0, x1 and x2 to be theperiod-3 points. We may assume without loss of generality that
x0 < x1 < x2.
(otherwise the points have the ordering x0 < x2 < x1 and in what follows we may re-define I0 =[x2, x1] and I[x0, x2] and the rest of the proof remains unchanged.)
We define the two intervals I0 = [x0, x1] and I1 = [x1, x2]. Since f(x0) = x1, f(x1) = x2 and f iscontinuous, then
f(I0) ⊃ I1. (54)
Similarly, since f(x1) = x2, f(x2) = x0 and f is continuous, then
f(I1) ⊃ I0 ∪ I1.
Hence, it follows that
f(I1) ⊃ I0, (55)
f(I1) ⊃ I1. (56)
We now construct periodic orbits of all periods:
• Period 1: From (56) and Lemma 1, we can conclude that f has a fixed point in I1.
• Period 2: Applying Lemma 2 to (55) we can infer the existence of an interval I ′1 ⊂ I1 suchthat f(I ′1) = I0. Now from (54), we have
f(f(I ′1)) = f(I0) ⊃ I1 ⊃ I ′1,
and now we can apply Lemma 1 to find that f2 has a fixed point in I ′1 (and hence a period-2orbit of f). Since f(I ′1) = I0, then the other point of the period-2 orbit is in I0 and so thereis a non-trivial period-2 orbit that oscillates between the intervals I0 and I1.
• Period n > 3: The idea is to use Lemma 2 n − 2 times to generate a nested set of n − 2subintervals of I1 all of which contain a fixed point of f . In order to complete the construction,we need two more intervals An−1 and An; see Figure below.
x1x0 x2
An-2
A2
A1
AnAn-1
54
Using Lemma 2 with (56), we can infer the existence of an interval A1 ⊂ I1 such that f(A1) =I1. Since A1 ⊂ I1 and f(A1) = I1 ⊃ A1, we can use Lemma 2 again to find another intervalA2 ⊂ A1 such that f(A2) = A1. Continuing this process n− 2 times, we generate a collectionof intervals Ai, i = 1, 2, . . . , n− 2 satisfying
An−2 ⊂ An−3 ⊂ · · · ⊂ A2 ⊂ A1 ⊂ I1,
withf(Ai) = Ai−1, i = 2, . . . , n− 2.
It can be shown that all these intervals contain a fixed point of f . Furthermore, we have
fn−2(An−2) = I1.
Now we need two more intervals An−1 and An to complete the construction. From (54) wehave f(I0) ⊃ I1 and we also know that I1 ⊃ An−2. Hence, by Lemma 2, there exists an intervalAn−1 ⊂ I0 such that f(An−1) = An−2.
Finally, since f(I1) ⊃ I0 ⊃ An−1, there is an interval An ⊂ I1 such that f(An) = An−1. Wenow have n subintervals such that f(Ai) = Ai−1, i = 2, . . . , n and f(A1) = I1. Thus,
fn(An) = I1 ⊃ An,
and so by Corollary 1, f has a period-n point. It can be shown that the minimal period ofthis period-n point is n.
6.6 Lyapunov Exponents
The sensitive dependence on initial conditions property of chaotic systems describes how (arbitrarily)small differences in initial conditions get amplified as the iteration proceeds. A Lyapunov expo-nent characterises the average rate of growth of small differences. It can also be used to calculatestability of periodic orbits.
Lyapunov exponents can be defined for both ODEs and maps. We will just concentrate on Lyapunovexponents for maps.
The key idea is this: we consider an initial condition x0 and its orbit xn. We then consider a smallperturbation x0 + ε0 and its orbit xn + εn. Here, εn measures the separation between the two orbitsxn and xn + εn. If |εn| ≈ |ε0|enλ, then λ is called the Lyapunov exponent. A negative λ impliesthat |εn| → 0 and the orbits converge. On the other hand, a positive Lyapunov exponent tells usthe orbits diverge and is a signature of chaotic dynamics.
To derive a formula for computing λ for a map xn+1 = f(xn), where f(xn) is a continuous anddifferentiable function, we start with the relation
|εn| ≈ |ε0|enλ,
and rearranging for λ, taking logs of both sides, we arrive at
λ ≈ 1
nln
∣∣∣∣εnε0
∣∣∣∣ .
55
Using the fact the separation between the two orbits εn = fn(x0 + ε0)− fn(x0), we find
λ ≈ 1
nln
∣∣∣∣εnε0
∣∣∣∣
=1
nln
∣∣∣∣fn(x0 + ε0)− fn(x0)
ε0
∣∣∣∣
=1
nln
∣∣∣∣d
dx(fn(x0))
∣∣∣∣ , (taking the limit as ε0 → 0),
=1
nln
∣∣∣∣∣n−1∏
k=0
d
dxf(xk)
∣∣∣∣∣ , (using chain rule on (fn(x0))′ )
=1
n
n−1∑
k=0
ln |f ′(xk)| , (replacing the log of a product with the sum of logs).
Definition 14 (Lyapunov Exponent) The Lyapunov exponent of a 1D map xn+1 = f(xn) isdefined as
λ = limn→∞
1
n
n−1∑
k=0
ln |f ′(xk)| , (57)
if the limit exists.
We can use the Lyapunov exponent to tell us about stability of periodic orbits as well as sensitivedependence on initial conditions
Example 35: Condition for stable periodic orbits Suppose that x0 is a period m point sothat xm = x0. Then the Lyapunov exponent becomes
λ =1
m
m−1∑
k=0
ln |f ′(xk)| .
The periodic orbit is stable if λ < 0 i.e., |εn| = |ε0|enλ → 0. Hence a periodic orbit is stable if
m−1∏
k=0
|f ′(xk)| < 1.
Example 36: Lyapunov exponent of the Logistic map In general, one needs to numericallycalculate Lyapunov exponents. We do this for the Logistic map xn+1 = rxn(1 − xn). We fix r andcompute, for a random initial condition, the first 300 iterates of the Logistic map (note we need onlystore the current iterate). We then compute the next 10,000 iterates of the Logistic map, and foreach iterate we compute ln |r− 2rxn| and add this to a running of logs. Finally, we divide the totalsum of logs by 10,000. Carrying out this procedure for a range of values of r, yields the Figure 30.
We see the Lyapunov exponent is negative for the stable periodic orbits. At each period-doublingbifurcation, the Lyapunov exponent becomes zero, indicating a bifurcation. In the chaotic region,we see the Lyapunov exponent is positive with interspersed dips in the periodic windows.
These dips occur when the periodic orbit of the Logistic map has each iterate xi with f ′(xk) = 0i.e., the Lyapunov exponent is ln(0) = −∞. When this occurs, the orbit is said to be super-stable.Orbits that are super-stable have one of the iterates is xk = 1
2 so that f(1/2) is at the maximum ofthe map.
56
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4- 1
- 0.8
- 0.6
- 0.4
- 0.2
0
0.2
0.4
0.6
0.8
1
Lyapunov exponent
r
period doubling bifurcations
chaos
2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.00.0
0.2
0.4
0.6
0.8
1.0
x
x = 1 − 1
r
period doubling bifurcations
xp+
xp−
chaos
r
periodic window
Figure 30: Plot of the Lyapunov exponent of the Logistic map as r is varied. Matlab code: [Lyapunovexponent Logistic map]
6.7 Universality and Re-normalisation
Qualitatively, the same period-doubling sequence is observed in a variety of maps and differentialequations e.g., the sine map
xn+1 = r sin(πxn), r, xn ∈ R,
and the Rossler system
x = −y − z,y = x+ ay,
z = b+ z(x− c), (x, y, z) ∈ R3, a, b, c ∈ R.
The bifurcation diagrams for both the Rossler system and the sine map are shown in the figuresbelow.
For the sine map, we see the bifurcation diagram looks amazingly similar to that of the Logisticmap shown in Figure 27 except the bifurcation parameter now runs from 0 to 1 rather than 0 to4. The scaling of the bifurcation diagram by a factor of 4 is impart, due to the sine map having amaximum of r (at x = 1
2 ) whereas the Logistic map has a maximum of 14r (also at x = 1
2 ). However,the similarity is only qualitative! In the sine map, the period doubling bifurcations occur earlier andthe periodic windows are narrower.
57
0.7 1
1
x
r
Sine map
xmax(N)
c
Rossler system
4 185
30
period-3 windowperiod-3 window
period-doubling
period-doubling
Figure 31: Bifurcation diagrams for the Rossler system and the sine map xn+1 = r sin(πxn). Matlab codesfor generating the bifurcation diagram for the Rossler system can be found here and for the sine map here.Matlab codes: [Bifurcation diagram: Rossler] and [Bifurcation diagram: Sine map]
The bifurcation diagram for the Rossler system is also amazingly similar to that of the Logistic mapwith period-doubling bifurcations and periodic windows (in particular the famous period-3 windowalso seen in the Logistic map and sine map!).
What is making the Rossler system, sine map and Logistic map behave similarly? They are eitherone-humped, unimodal maps (in the case of sine and Logistic map) or the dynamics can be reducedto approximately a unimodal map (in the case of the Rossler system; see Figure 23).
Definition 15 (Unimodal maps) Unimodal maps have a single maximum, smooth and concavedown.
In particular, if a unimodal map satisfies the following conditions then the map xn+1 = rf(xn) willundergo the period-doubling route to chaos
1. f(0) = f(1) = 0,
2. f is a smooth function which has a quadratic maximum at xm, i.e., f ′′(xm) 6= 0,
3. f is monotonic in the intervals [0, xm) and (xm, 1],
4. f has a negative Schwarzian derivative i.e.,
f ′′′(x)
f ′(x)− 3
2
(f ′′(x)
f ′(x)
)2
< 0.
For the discussion that follows, we consider a unimodal map f(x, r) that undergoes a period-doublingroute to chaos as r is increased, with the maximum value of f occuring at xm (in the Logistic mapand sine map, xm = 1
2 ). Furthermore, we denote rn to be the value of r where a 2n-cycle is bornand Rn is the r value where the 2n-cycle is super-stable i.e., each iterate xi of the periodic orbit hasf ′(xi) = 0.
Example 37: Super-stable orbits in the Logistic map Fixed points (period-1 orbits) of theLogistic map
xn+1 = rxn(1− xn),
are given by x = 0 and x = 1− 1r . Looking at the Floquet multipliers of the fixed points shows that
fx(0, r) = 0 at r = 0 and fx(1 − 1r , r) = 0 at r = 2. Hence, the fixed point x = 0 is super-stable
when r = 0 and the fixed point x = 1− 1r is super-stable at r = 2 i.e., R0 = 2.
At r = 3, we found that the fixed point x = 1− 1r underwent a period-doubling bifurcation where a
period-2 cycle was born
x± =r + 1
2r± 1
2r
√(r − 3)(r + 1).
58
In the section on Lyapunov exponents, we saw that a periodic orbit is stable if
m−1∏
k=0
|f ′(xk)| < 1,
where x0 = xm. Hence, the period-2 cycle is super-stable if
fx(x+) · fx(x−) = 0,
i.e.,
fx(x+) · fx(x−) = r(1− 2x+)r(1− 2x−),
= r2[1− 2(x+ + x−) + 4x+x−],
= r2[1− 2(r + 1)/r + 4(r + 1)/r2],
= 4 + 2r − r2 = 0, when r = 1 +√
5.
Hence, the period-2 cycle is super-stable when r = 1 +√
5 i.e., R1 = 1 +√
5.
Note, that the period-2 cycle undergoes a period doubling bifurcation at r = 1 +√
6.
General rule for unimodal maps: a period-2n cycle is born in a period-doubling bifurcation,becomes super-stable (hence the dips in Figure 57), and then becomes unstable in a period-doublingbifurcation; where the process is repeated but for the now stable period-2n+1 cycle. The super-stablecycle always contains xm as one of its points.
Graphically, the location of the super-stable 2n-cycles can be found by looking at where the liney = xm intersects with the bifurcation curves in the following Fig-tree diagram. If successive rn
r
x
xm
R0 R1 R2r1 r2 r3
(the period-doubling bifurcation points) are shrinking by the universal factor δ ≈ 4.669, then so doRn (the super-stable points).
The key idea for showing that the ratio between successive Rn
δ = limn→∞
Rn −Rn−1Rn+1 −Rn
= 4.6692 . . .
is a universal constant, is the self-similarity of the fig-tree diagram: The branches look the sameas those ‘higher’ up the tree, just scaled in both r and x. It is this similarity that leads to theself-similar period-doubling sequence in the periodic windows described in §6.4.
To understand this self-similarity better, lets look at the graphs of y = f(x, r) at r = R0 andy = f2(x, r) at r = R1. We then want to transform, (re-normalize), one map into the other bya scaling in r and x. This seems like a good idea since the fixed point xm of both the maps, aresuper-stable.
We take the Logistic map as a specific example just for the moment and plot f(x,R0) and f(x,R1).We see that around xm = 1
2 , the graphs look qualitatively the same if we employ a flip of the
59
xm
flip
xm
f(x, R0) f2(x, R1)
f2(x,R1)-graph near xm.
Our first step in the analysis of this transformation is a shifting xm to the origin by letting x = x+xmhence
xn+1 = f(xn, r) ⇒ xn+1 = g(xn, r).
Now, the function g(x, r) := f(x− xm, r)− xm has a maximum at x = 0 where g(0, r) = 0. We nowplot the same transformation shown in the figures above but to g. We can make the middle figure
iterate rescale byα = −2.5 . . .
0 00
g(x, R0)
g2(x, R1)αg2
(x
α, R1
)
look like the first if flip the figure (i.e., (x, y)→ (−x,−y)) and then ‘blow it up’ by a factor |α| > 1.These two transformations can be done in one go if we choose the scale factor α to be negative. Soto renormalize g, we take its second iterate, rescale x→ x/α, and shift r to the next super-stableparameter value i.e.,
g(x, R0) ≈ αg2(x
α,R1
).
For each super-stable parameter value, we can carry out the same renormalization procedure (ong2(xα , R1
)) n times to find
g(x, R0) ≈ αng(2n)(x
αn, Rn
).
If α = −2.5029 . . . is chosen correctly, then in the limit as n→∞ it is found
limn→∞
αng(2n)
(x
αn, Rn
)= h0 (x),
where h0 is a universal function with a super-stable fixed point at the origin x = 0. The limitonly exists if α is chosen correctly.
The special thing about universal functions is that they only see the (local) maximum of g and theydo not depend on the global information of g. This occurs since x/αn → 0 as n → ∞ and so h0only depends on g through its behaviour near x = 0.
However, h0 does depend on the order of the maximum of g, that is h0 is the same (universal) forall quadratic-maximum maps (maps with fxx(xm) 6= 0). A different universal h0 is found for allquartic-maximum map i.e., maps with fxx(xm) = 0, fxxxx(xm) 6= 0, though this is not generic.
We can find other universal functions hi by starting with g(x, Ri) instead of g(x, R0) to find
hi(x) = limn→∞
αng(2n)
(x
αn, Rn+i
),
where hi is a the universal function with a super-stable 2i-cycle.
60
The really important case is the universal function h∞ with R∞ since
g(x, R∞) ≈ αg2(x
α,R∞
),
and we see for once we don’t have to shift R when we renormalise! The limiting universal functionh∞(x), usually called h(x), satisfies the functional equation
h(x) = αh2(x
α
). (58)
Solving this functional equation requires finding the universal scale factor α and the universal func-tion h(x). This is a hard equation to solve!
Functional equations are like differential equations and to solve them we need to define some bound-ary conditions. Since our shifted map g(x) has a maximum at x we require hx(0) = 0. Also, we canset h(0) = 1 without loss of generality (This just defines the scale for x; if h(x) is a solution of (58)then so is βh(x/β), with the same α).
At x = 0 the functional equation yields h(0) = αh(h(0)), substituting in the boundary conditionh(0) = 1, we find
α =1
h(1).
Hence, α is determined by g at x = 1. Now the hard part. . . finding a closed form solution of g; sofar no one has managed it. Feigenbaum resorted to be power series approximation
h(x) = 1 + a2x2 + a4x
4 + · · · ,
which assumes the maximum of g is quadratic. The coefficients a2, a4, . . . are found substituting thepower series into (58) and equating coefficients for like powers of x. Taking seven terms in the powerseries approximation, we find a2 ≈ −1.5276, a4 ≈ 0.1048. Evaluating h(1) using these coefficients,we find the scale factor α ≈ −2.5029.
This renormalisation theory can also give us the Feigenbaum constant δ = 4.6692 . . . but this requiresa lot of Functional Analysis; see third year Introduction to Function spaces (MAT3004)!
Instead, we will follow the algebraic calculations done in chapter 10.7, Strogatz’s book, NonlinearDynamics & Chaos [5]. We start with a unimodal map f(x, r) that undergoes an infinite sequenceof period-doubling bifurcations. From Theorem 6.4 we know near each period-doubling bifurcation,the unimodal map is topologically equivalent to the normal form (53). However, we wish to considera slightly different normal form
yn+1 = f(yn, λ) := −(1 + λ)yn + y2n,
where we have ignored the positive cubic term, y3n in (53). Algebraically, all maps near a period-doubling bifurcation have this form.
Now, we know for λ > 0, there exist a period-2 cycle x = p and x = q such that
f(p, λ) = q, f(q, λ) = p,
i.e., f2(p, λ) = f(f(p)) = p and f2(q, λ) = f(f(q)) = q. We can solve for p and q to find
p =λ+√λ2 + 4λ
2, q =
λ−√λ2 + 4λ
2.
Note that p is a fixed point of f2.
We now shift the origin to p and look at the local dynamics similar to the first renormalisation stepin shifting the location of the super-stable points Ri. To do this, we expand
p+ ηn+1 = f2(p+ ηn, λ),
in powers of the small perturbation ηn. After a bit of algebra and neglecting higher order terms, wefind
ηn+1 = (1− 4λ− λ2)ηn + Cη2n + . . . (59)
61
whereC = 4λ+ λ2 − 3
√λ2 + 4λ. (60)
Algebraically, the map (59) has the same form as the period-doubling normal form (53). To turnit into exactly the same as the period-doubling normal form, we need to carry out the secondrenormalisation step; rescaling η. To do this, let yn = Cηn, we (59) becomes
yn+1 = (1− 4λ− λ2)yn + y2n + . . .
and we define a new parameter λ such that −(1 + λ) = (1− 4λ− λ2) then we get
yn+1 = −(1 + λ)yn + y2n, (61)
which is exactly the same as the period-doubling normal form. Note, C plays the role of the rescalingfactor α in the renormalisation theory.
Now, when λ = 0, the renormalised map (61) undergoes a period-doubling bifurcation correspondingto the birth of a period-4 cycle in the original unimodal map f .
Solving λ = 0, we find λ = −2 +√
6. This value predicts the creation of the period-4 orbit in theLogistic map since at r = r1 = 3 (the first period-doubling bifurcation) corresponds to λ = 0, hencer2 = 3 + (−2 +
√6) = 1 +
√6 (the location of the second period-doubling bifurcation in the Logistic
map!)
Since (61) has the same form as the period-doubling normal form, we can carry out the samerenormalisation transformation over and over again, until the onset of chaos!
Let λk be the bifurcation value where the original map creates a period-2k orbit. So far we havefound λ1 = 0 and λ2 = −2 +
√6. In general, λk satisfy
λk−1 = µ2k + 4λk − 2.
Solving for λk we getλk = −2 +
√6 + λk−1.
This is a 1D map for the period-doubling bifurcation points with λ1 = 0. This map has a stablefixed point λ∗ given by
λ∗ =1
2
(−3 +
√17)≈ 0.56.
For the logistic map r∞ ≈ 3+λ∗ ≈ 3.56, whereas the actual value is r∞ ≈ 3.57! Not bad consideringthe massive approximations being made.
Recall that the Feigenbaum constant δ is
δ = limk→∞
rk − rk−1rk+1 − rk
≈ limk→∞
λk−1 − λ∗λk − λ∗
.
Since this ratio tends to 0/0 as k →∞, we may use L’Hopital’s rule to find
δ ≈ dλk−1dλk
∣∣∣∣λ=λ∗
= 2λ∗ + 4 = 1 +√
17 ≈ 5.12.
This approximation is within 10% of the true value δ = 4.6692. Substituting λ∗ into C, we findC ≈ −2.24 which is also within 10% of the true value α = −2.5029.
62
7 Chaotic 1D maps
The logistic map isn’t the only 1D map that exhibits chaotic dynamics. Two simpler examples arethe Doubling map and the Tent map.
7.1 The Doubling map
The Doubling map is defined as
xn+1 = D(xn) =
{2x if x ∈ [0, 12 ),2x− 1 if x ∈ [ 12 , 1]
(62)
Here the interval [0, 1] gets stretched to twice its length, cut in half, and then each half is mappedonto the interval [0, 1]. This is a discontinuous version of the “stretch and fold” mechanism for chaos.This is similar to rolling out pastry, cut it in half, and overlay it.
1
0
x
0 10.5
0 1
x
0 2
0 1 2
1
overlay
0 1
2
7.2 The Tent map
The tent map is defined as
xn+1 = T (xn) =
{2x if x ∈ [0, 12 ),2− 2x if x ∈ [ 12 , 1]
(63)
Here the interval [0, 1] gets stretched to twice its length, folded in half, and then each half is mappedonto the interval [0, 1]. This may is continuous but not differentiable at x = 1
2 . It is called the Tentmap since the graph looks like a tent.
1
0
x
0 10.5
0 1
x
0 2fold
0 1
7.3 Symbolic Dynamics
Let us start by looking at the Doubling map. We wish to understand the orbits of the Doublingmap (some orbits are boring! e.g, x0 = 0. We will ignore these). Our aim is to somehow link theorbits of the Doubling map to a simpler map that we can analyse.
63
To do this we will write down a “0” every time an iterate xi lands in the interval [0, 12 ) and a “1”every time an iterate lands in the interval [ 12 , 1]. Now we can write down a sequence of 0’s and 1’sthat represent a trajectory as
sD(x0) = .a0a1a2, . . . , ai ∈ {0, 1},
where
ai =
{0 if Di(x0) ∈ [0, 12 ),1 if Di(x0) ∈ [ 12 , 1]
We call the sequence sD(x0) the itinerary of x0. For example, if we consider the iterates of x0 = 0.1under the Doubling map, we get
0.1, 0.2, 0.4, 0.8, 0.6, 0.2, 0.4, . . .
Then its itinerary issD(0.1) = .001100 . . .
since x0 = 0.1 ∈ [0, 12 ), hence a0 = 0, 0.4 ∈ [0, 12 ) ≡ a1 = 0 and 0.8 ∈ [ 12 , 1] ≡ a2 = 1, etc.
The space of one-sided sequences (i.e., sequences that extent infinitely long in one direction) calledthe sequence space. We define the sequence space on two symbols (0 and 1) as follows
∑
2
= {s = .a0a1a2 . . . | ai = 0 or1} = {0, 1}N
We have specified a method to transform an orbit of the Doubling map to an itinerary in the space∑2. This can be defined as the map sD : [0, 1]→∑
2
x 7→ sD(x).
This map is invertible i.e., for each xi ∈ [0, 1] there is exactly one ai ∈∑
2. For the Doubling map,we can write down explicitly the inverse map s−1D
s−1D (.a0a1a2 . . .) =1
2
∑ ai2i∈ [0, 1].
Hence, sD associates the binary expansion to x ∈ [0, 1]. Another way to see this is by the fact thatthe Doubling map is actually the map f(x) = 2x(mod1).
We wish the space of itineraries∑
2 to act like the dynamics of the Doubling map. To do this wewill define the shift map σ :
∑2 →
∑2 where
σ(.a0a1a2a3 . . .) = .a1a2a4 . . . ,
so σ “forgets” the first symbol of the sequence a0. Since a0 may be either 0 or 1, σ is a two-to-one map of
∑2. The Doubling map D behaves on [0, 1] exactly like the shift map σ on
∑2 i.e.,
D = s−1D ◦ σ ◦ sD and is represented by the diagram
D[0, 1] −→ [0, 1]
sD ↓ ↓ sD∑
2 −→ ∑2
σ
So to understand the dynamics of the Doubling map, we now need to just understand the dynamicson∑
2 which is easier. First though we need some preliminary results.
The space∑
2 is a metric space with distance d
d(s, t) =
∞∑
i=0
|si − ti|2i
64
between sequences s = .a0a1a2 . . . and t = .b0b1b2 . . .
Since |si − ti| is either 0 or 1, this sequence is dominated by the geometric series
∞∑
i=0
1
2i= 2,
and therefore it converges (reassuring to know!).
Example 38: If s = .0000 . . . and t = .1111 . . . then d(s, t) = 2. If r = .1010 . . ., then
d(s, r) =
∞∑
j=0
1
2j=
∞∑
i=0
1
22i=
1
1− 14
=4
3,
where j = 2i.
The metric d allows us to decide which subsets of∑
2 are open and which are closed, as well aswhich sequences that are close to each other.
Lemma 3 Let s, t ∈∑2 and suppose si = ti for each i = 0, 1, . . . , n. Then
d(s, t) ≤ 1
2n.
Conversely, if d(s, t) < 12n , then si = ti for i ≤ n.
Proof. If si = ti for i ≤ n, then
d(s, t) =
n∑
i=0
|si − si|2i
+
∞∑
i=n+1
|si − ti|2i
,
≤∞∑
i=n+1
1
2i=
1
2n.
On the other hand, if sj 6= tj for some j ≤ n, then we must have
d(s, t) ≥ 1
2j≥ 1
2n.
Consequently, if d(s, t) < 12n , then si = ti for i ≤ n.
This result allows us to ascertain whether or not two sequences are close to each other. In otherwords, two sequences are close to each other in
∑2 provided their first few entries agree.
Theorem 7.1 (Chaotic dynamics of σ) The dynamical system on∑
2 defined by a 7→ σ(a) sat-isfies the following properties:
1. σ has ∞ many periodic orbits,
2. σ has ∞ many non-periodic orbits,
3. σ has dense periodic points
4. σ has a dense orbit, that is, orbits that come arbitrarily close to any point in∑
2
Proof.
1. Any sequence of the form s = .a0a1a2 . . . ak (repeated sequence) is periodic.
65
2. Consider the non-periodic orbit s0 = .01 001 0001 · · · 00 . . . 01, made up of blocks of sizen = 1, 2, . . . containing n− 1 zeros and a 1. Also σi(s0) is non-periodic for any i.
3. Dense periodic points: Let s = .a0a1a2 . . . be any initial condition. Now choose t = .a0a1a2 . . . akfor some k. Then by theorem 3, s and t differ by at most 1
2kand t is periodic. Thus we can
find a periodic point as close as we like to s by taking longer and longer periods.
4. We enumerate sequences of length n of all possible n combinations of 0, 1 i.e.,
n 1 2 3seq’s 0,1 00 10 01 11 000 001 · · ·s = .01 00100111 000001. . .
A dense orbit is constructed by applying σi to s such that it hops around almost everywherein∑
2 i.e. if you choose a point in s ∈∑2 and you wish d(s, σi(sd)) < 2−N = ε, then i = N .
As for the Doubling map, we can relate the dynamics of the tent map to the dynamics of the shiftmap σ on the space
∑2. Hence the tent map is also chaotic. In fact, we can relate most one hump
maps to the dynamics of the shift map on∑
2 including the logistic map
xn+1 = 4xn(1− xn).
We note the shift map on∑
2 has two other important properties:
1. Sensitive dependence on initial conditions. A map has sensitive dependence on initial condi-tions if, given any initial point x0, there is a point y0 arbitrarily close to x0 such that theiterates starting with x0 and y0 will eventually differ by an amount δ > 0 i.e., for some n
|xn − yn| ≥ δ.
Choose two orbits s, r ∈∑2 such that d(s, r) ≤ 2−n. This implies that the first n symbols areidentical and the first non-equal symbol occurs at n+ 1th term. We now consider
d(σn+1(s), σn+1(r)) = d(.an+1an+2 . . . , .bn+1bn+2 . . .) =
∞∑
i=0
|an+1+i − bn+1+i|2i
= 1 +
∞∑
k=1
|an+1+k − bn+1+k|2i
≥ 1.
2. Mixing. Given any two arbitrarily small intervals I and J . A map is mixing if there is someinitial point x0 ∈ I whose orbit enters the interval J after a number of iterations. It is thisproperty, that allows the Taffy machine to mix the saltwater taffy.
66
8 Fractals
Roughly speaking, a fractal is a complex geometric object that if you zoom-in, at arbitrarily smallscales, the object looks self-similar to that of the whole. There has been a lot of interest in fractalsbecause of their beauty, complexity and endless structure. Fractals seem to look approximately likenatural objects from snowflakes, blood vessel networks and lightening across the sky. In order theunderstand fractals, we need to move away from classical geometry and develop a theory for highlyirregular objects.
To see an introduction to Fractals, the Chaos & Fractals website has a link to a movie explainingsome the key properties of Fractals; [Fractals].
Let us begin with an example of a fractal.
Example 39: Cantor set In 1883, Georg Cantor published a paper on the construction of asimple set that had many peculiar properties. To construct it, Cantor started with the unit intervalC0 = [0, 1] and then removed the open interval ( 1
3 ,23 ) i.e., the middle third of the interval. This
leaves the two intervals of length 13 which are [0, 13 ] and [ 23 , 1]. The remaining set is the union
C1 = [0, 13 ] ∪ [ 23 , 1]. Repeating this procedure, infinitely many times, of removing the middle thirdof the remaining intervals yields the Cantor set C.
10 x
1
3
2
3
The Cantor set has the following fractal properties:
1. C has structure at arbitrarily small scales. Not matter how much we zoom-in and look atC, we see that it has points separated by gaps. This is most unlike usual structures (e.g., asquare) in that as you zoom-in the picture becomes more featureless (e.g, a line).
2. C has self-similar structure. As we zoom-in we see exactly the same copies of the whole. Note,most fractals don’t have this exact self-similarity.
3. C has non-integer dimension. In fact, as we say see it has dimension d = log 2/ log 3 ≈ 0.63.
The Cantor set also has some other interesting (non-fractal) properties
• C has zero length. To see this, we note that the length of C0 is 1, length of C1 is 23 , the length
of C2 is 49 etc. Hence, the length of Ck is (2/3)k. Taking the limit as k →∞ gives
length of C = limk→∞
(2
3
)k= 0.
• C has infinitely many points. See exercise sheet.
So C seems to belongs somewhere in between a set of points and a line.
Example 40: Koch curve The Koch curve was first described in 1904. It is constructed bystarting with a line of length 1. The middle third of this line is then replaced by two sides of anequilateral triangle. This gives four line segments. Repeating this procedure infinitely many timesleads to the Koch curve. Some interesting properties of the Koch curve:
67
step 0
step 1
step 2
step 3
Koch curve
Koch snowflake
Figure 32: Kock curve and Koch Snowflake.Matlab code: [Koch snowflake] and [kochstep.m]
1. The Koch curve has infinite length. To see this we note the initial line has length 1 and soat the first step, a line of length 1
3 is replaced by two line segments with combined length 23 .
Thus the length of the line at the first step is 43 . Similarly, at each step, the length increases
by a factor of 43 so that the length at step k is (4/3)k. Taking the limit as k →∞ yields
length of the Koch curve = limk→∞
(4
3
)k=∞.
So the Koch curve has infinite length but is contained within a finite area.
2. The Koch curve is self-similar. For example, taking the left hand quarter of the curve andmultiplying by 3 gives the original Koch curve again.
3. The Koch curve is continuous but nowhere differentiable.
Thus, the Koch curve seems to fit somewhere between a line and a two-dimensional area.
8.1 Self-similarity
For the purpose of this course, we will restrict our discussion of fractals to those that reside on theplane R2. One-dimensional fractals are obvious subsets in R2, while the discussion can be generalisedto fractals in Rn.
Let us first recall some basic definitions of sets in R2. We call a set in R2 bounded if it can be enclosedby a suitably large circle and closed if it contains all of its boundary points. Two sets in R2 arecongruent if they can be made to coincide exactly by translating and rotating them appropriatelywithin R2. If T : R2 → R2 is the linear operator (matrix) that scales by a factor s, and if S is a setin R2, then the set T (S) (the set of images of points in S under T ) is called a dilation of the set Sif s > 1 and a contraction of S if 0 < s < 1.
Definition 16 (Self-similarity) A closed and bounded subset of the plane R2 is said to be self-similar if it can be expressed in the form
S = S1 ∪ S2 ∪ S3 ∪ · · · ∪ Sk, (64)
where S1, S2, S3, . . . , Sk are nonoverlapping sets, each of which is congruent to S scaled by the samefactor s (0 < s < 1).
68
Bounded setEnclosing circle
x
y
x
y
Congruent sets
x
y
x
y
T
[xy
]=
[s 00 s
] [xy
]
T (S)
S
Contraction
Example 41: A square can be expressed as the union of four non-overlapping congruent squares.We have separated the four squares slightly so that they can be seen more easily. Each of the foursmaller squares is congruent to the original square scaled by a factor of 1
2 i.e., a square is a self-similarset with k = 4 and s = 1
2 .
8.2 Fractal Dimension
Intuitively, we may define the dimension of a subspace of a vector space to be the number of vectorsin a basis of the subspace. This definition is a special case of a more general concept called topologicaldimension (dT (S)). We wont give a precise definition here but informally we say
• a point in R2 has topological dimension zero,
• a curve in R2 has topological dimension one,
• a region in R2 has topological dimension two.
An alternative definition for the dimension of an arbitrary set in Rn was given by the Germanmathematician, Felix Similarity in 1919. His definition is quite complicated, but for self-similar setsit reduces to something more simple
Definition 17 (Similarity dimension) The Similarity dimension of a self-similar set S of theform (64) is denoted by dsim(S) and is defined by
dsim(S) =ln k
ln(1/s)(65)
69
There are a couple of things to note about the Similarity dimension
• The topological dimension and Similarity dimension of a set need not be the same.
• The Similarity dimension of a set need not be an integer.
• The topological dimension of a set is less that or equal to the Similarity dimension i.e., dT (S) ≤dsim.
Example 42: Cantor set Starting from the line segment (interval) C0 = [0, 1], the first step ofthe construction of the Cantor set, C1 can be thought of as the scaling C0 by a scale factor 1
3 to getS1 = [0, 13 ] and adding a copy of S1 shifted by 1
3 i.e., S2 = S1 + 23 = [ 23 , 1]. Therefore C1 = S1 ∪ S2
and s = 13 , k = 2.
We can now compute the Similarity dimension of the Cantor set
dsim(C) =ln 2
ln(3)≈ 0.63
Example 43: Koch curve The construction of the Koch curve can be thought of the compositionof four line segments, each scaled down by a factor of 1
3 of the original interval i.e., s = 13 , k = 4.
Hence, the Similarity dimension of the Koch curve is
dsim(K) =ln 4
ln(3)≈ 1.26,
confirming our first impression that the Koch curve was somewhere between a line and a two-dimensional area.
We note there are other ways of measuring the dimension of objects in Euclidean space with themost popular being box dimension, pointwise dimension and correlation dimension. These methodsallow us to cope with fractals that are not strictly self-similar. We will only deal with the boxdimension.
The box dimension is based on covering an object with boxes which have sides of length h, ignoringfeatures that occur as scales less than h, and then counting the number of boxes N(h) as h→ 0.
hh
As h is reduced, N(h) increases and we assume a power law relationship between N(h) and 1/h
N(h) =c
hd,
where c is a fixed constant. Taking logs and rearranging, we find
d =ln(N(h))
ln(1/h)− ln(c)
ln(1/h).
Now as h→ 0, the last term vanishes and we are left with the Box dimension.
Definition 18 (Box Dimension) The Box dimension of a set S is denoted dB and is defined by
dB = limh→0
ln(N(h))
ln(1/h),
if the limit exists.
70
Example 44: Box Dimension of a non self-similar fractal We construct a non self-similarfractal as follows. A square region is divided into nine squares, and then two of the small squaresare selected at random and discarded. Then the process is repeated on each of the remaining sevensmall squares, and so on. We pick the length of the original square to be equal to one. Then S1
S1 S2
is covered by N = 7 squares of sides h = 13 . Similarly, S2 is covered by N = 72 squares of side
h =(13
)2. In general, N = 7n when h =
(13
)n. Hence,
dB = limh→0
ln(N(h))
ln(1/h)=
ln(7n)
ln(3n)=n
n
ln 7
ln 3=
ln 7
ln 3.
Definition 19 (Fractal) A fractal is a subset of a Euclidean space whose (Box/Similarity)-dimensionis non-integer.
Example 45: Cantor-like set in the period doubling route to chaos:
Consider the logistic map xn+1 = rxn(1 − xn) at r = r∞ = 3.5699456, . . ., corresponding to theonset of chaos. We can visualise the attractor by building it up recursively. At each period doublingbifurcation n, we have a 2n periodic orbit. The dots in the bifurcation diagram on the left, correspond
r
x
r=3.5699...
chaos
to stable 2n-periodic orbits with the right panel showing the corresponding x values of the dots. Asn→∞, the set of points in the right panel approaches a Cantor set. However, this set isn’t strictly aCantor set since it isn’t self-similar as the gaps scale by different factors depending on their location.The resulting set is called a topological Cantor set.
Definition 20 (Topological Cantor Set) A closed set S is called a topological Cantor set if itsatisfies the following properties:
71
1. S is “totally disconnected” i.e., S contains no connected subsets (other than single points).Informally, it means all points are separated.
2. S contains no “isolated points” i.e., every point in S has a neighbour arbitrarily close by.
We note these properties contrast each other since the first property says points in S are spread apartwhile the second property says they are all packed together. Furthermore, a topological Cantor setis not required to be self-similar or dimension.
The box dimension for this topological Cantor set at the onset of chaos in the Logistic map has beenfound to be roughly dB ∼ 0.538. This is an example of a strange attractor.
72
9 Strange Attractors and Repellers
9.1 Attractors
So far we have seen three examples of a chaotic attractor, the Logistic map, the sine map, and theRossler system. In all of these cases, orbits of the dynamical systems converge to an attractor.
Definition 21 (Attractor) An attractor is a closed set A with the following properties:
1. A is an invariant set: any trajectory x(t) or xn that starts in A stays in A for all time.
2. A attracts an open set of initial conditions: there is an open set U , containing A, such thatif x(0) ∈ U or x0 ∈ U , then the distance from x(t) or xn to A tends to zero as t → ∞. Thelargest U is called the basin of attraction of A.
3. A is minimal: there is no proper subset of A that satisfies conditions 1 and 2.
The key thing to note here is that an attractor is bounded i.e., there are no trajectories that skipoff to infinity. However, chaotic attractors exhibit sensitive dependence on initial conditions wheretwo slightly different initial conditions diverge exponentially fast yet they remain bounded. It is theStretching-and-folding mechanism described in §7 that allows for both the sensitive dependenceon initial conditions and boundedness of trajectories.
Example 46: Rossler Attractor If we take a Poincare section of the Chaotic attractor of theRossler system, we slice through the all the folding of the attractor. Below, we sketch the Poincaresection of the attractor.
Poincare section
Lorenz section flatten and stretch
fold
repeat
If we take a further one-dimensional slice (known as a Lorenz section) through the Poincare section,we see an infinite set of intersections (points) with the Lorenz section. These points are separatedby gaps of various sizes. This set of points is a topological Cantor set. Hence, the Rossler attractorhas a non-integer (fractal) dimension.
Definition 22 (Strange Attractor) A strange attractor is an attractor of a dynamical systemwith a non-integer (fractal) dimension.
Example 47: The Taffy machine A physical example of a strange attractor is the Taffy machine;see the movie on the Chaos & Fractals website [Taffy machine]. The topology of this machine isvery complicated. The website:
[http://www.math.umd.edu/ halbert/road-show/Start-Here.html]
describes the topological of the Taffy machine using animations, without which understanding theaction of Taffy machine would be almost impossible!
73
The action of the Taffy machine can be described (approximately) by the 1D map Taffy map T :[0, 1] 7→ [0, 1]
xn+1 = T (xn) =
axn for 0 ≤ x < x1,2− axn for x1 ≤ x < x2,
axn −√
2 for x2 ≤ x < x3,
2 +√
2− axn for x3 ≤ x < 12
−T (1− xn) for 12 ≤ x ≤ 1
(66)
where a = 3 + 2√
2 is the constant stretch rate of the Taffy machine and x1 = 3 − 2√
2, x2 =1−√
2/2, x3 =√
2− 1.
It can be shown that this map has∞-many periodic and non-periodic orbits and the periodic pointsare dense on the unit interval; similar to Theorem 7.1.
There is also a 2D map that describes the Taffy machine; see the website above. This map can beshown to have a strange attractor.
9.2 2D maps
Example 48: The Baker map The Baker map of the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 to itselfis defined as
(xn+1, yn+1) = B(xn, yn) :=
{(2xn, ayn) for 0 ≤ xn < 1
2(2xn − 1, ayn + 1
2 ) for 12 ≤ xn ≤ 1,
(67)
where a ∈ (0, 12 ] is a parameter. This map is similar to the 1D Doubling map in the x-direction(shown in §7). If we start with the university of surrey logo on the unit square, the Baker mapcarries out the following steps:
1. the logo in the x-direction is stretched to x ∈ [0, 2] and shrunk in y-direction by a factor of a.
2. Then the rectangle is cut in half and yielding two 1 × a rectangles, which are stacked on topof each other.
For a < 12 , the Baker map has a fractal attractor A that attracts all orbits. To see what this attractor
looks like, we start with the unit square. This contains all possible initial conditions. Applying theBaker map to the unit square yields the following figure. We see that the set Bn(S) consists of2n-horizontal strips of height an. The limiting set A = B∞(S) is a fractal. Topologically, the fractalattractor is a Cantor set of line segments.
We can calculate the box-dimension of this fractal attractor, by looking at the covering of Bn withsquare boxes of side h = an. Since the strips have unit length, we need a−n boxes to cover each stripof Bn. There are 2n strips, implying the number of boxes to cover Bn is N ≈ a−n × 2n = (a/2)−n.Plugging this into the box-dimension formula yields
dB = limh→0
lnN(h)
ln(1h
) = limn→∞
ln[(a/2)−n]
ln(a−n)= 1 +
ln 12
ln a.
74
1
2
1
1
0
0
a
2
stretch and flatten cut and stack
S B(S) B2(S) B3(S)1
1
0
a +1
2
0
a
1
2
For a < 12 , the dimension of the Baker attractor is non-integer and hence it is a strange attractor.
One can carry out a symbolic dynamics analysis of the Baker map and prove a similar theorem asTheorem 7.1.
Example 49: Smale’s Horsehoe map Smale’s Horseshoe (H) map crops up in the analysis oftransient chaos explored in § 5.5.1. The Smale Horsehoe map does not have a strange attractor buta strange saddle set that remains the same under the action of the map.
The map is constructed as follows. We start with a unit square, stretch and flatten it. Taking thisflattened and stretched square, we fold it into a horsehoe and overlay it on the unit square andslice off the overhanging parts. The process is repeated with the new square. Most points in thesquare are eventually removed. In the context of transient chaos in the Lorenz equations, initialpoints starting in the square are stretched and folded creating chaotic dynamics. However, almostall orbits get mapped out of the square (they end up in the ‘overhangs’) and the orbits escape tosame distant part of phase space; for instance, a stable equilibrium in the Lorenz equations.
The remaining set of points that remain in the square (called an invariant set) under the forwarditeration of the horsehoe map can be shown to be a vertical Cantor set i.e., the set of points thatremain in the square under forward iteration of the Horseshoe map
Λ+ = {x ∈ S | Hk(x) ∈ S for k = 0, 1, 2, . . .},is a Cantor set. Similarly, the invariant set under the backward iteration of the horsehoe map can beshown to be a horizontal Cantor set i.e., the set of points that remain in the square under backwarditeration of the Horseshoe map
Λ− = {x ∈ S | H−k(x) ∈ S for k = 1, 2, 3, . . .},is a Cantor set. The dynamics on the Horseshoe
Λ = Λ+ ∩ Λ−,
75
S
a
a a
b
bb
c
c c
d
dd
b′
d′
b′d′
a′
c′
a′c′
flatten and stretch
fold
slice off the overhanging parts
re-inject
can be described using symbolic dynamics. In particular, any point (x) in Λ can be described by adoubly-infinite sequence of 0’s and 1’s
SH(x) = (. . . a−2a−1 · a0a1a2 . . .).
The dynamics on the Horseshoe is topologically conjugate to the shift map on the space of twosymbols. We have seen in Theorem 7.1, that this map has an infinite number of periodic andaperiodic points and correspondingly complicated (chaotic) dynamics.
One can show that any non-degenerate crossing of stable and unstable manifolds in any dynamicalsystem leads to a horseshoe map and correspondingly chaos.
76
10 Application: Secret communication with chaos
Chaos in deterministic systems means that the system behaves unpredictably. In other words, eventhough we know the rules by which the system evolves, we can’t predict the outcome. What use isthis knowledge?
One application has emerged utilising the idea of chaos to mask secret communications. Chaossounds a lot like noise and so a transmission masked by chaos would also sound like noise, remainingundetected by an outside listener.
The idea is as follows:
1. A chaotic signal u(t) is generated from an electronic realisation of a chaotic system (in thiscase the Lorenz equations) and added to a message m(t) which is then transmitted. The sizeof the chaotic signal is very much large than the message so as to make it sound as much aspossible like noise i.e., |m| � |u|.
2. The sent signal is received and feed into the same electronic chaotic system. Amazingly, thischaotic system then synchronises with the original chaotic system (at the transmitter end)producing a chaotic signal that can be subtracted from the received transmission to yield theoriginal message!
messagem(t)
a component of signal from chaotic circuit
u(t)
m<<u
u(t)+m(t)
transmitter
receiver
u(t)+m(t)drives identical circuit ur(t)
u(t)+m(t)-ur(t)
m(t)
All works via u(t)~ur(t) Chaotic Synchronisation
On the Chaos & Fractals website we have a movie demonstrating this idea; see [Sending Secretcommunications with Chaos]. Steven Strogatz’s book Nonlinear Dynamics and Chaos, providesfurther explanation.
So the key step in this method of communication is the synchronisation of two chaotic systems.Intuitively, you might think this is impossible for chaotic systems since they are highly sensitive toslight changes in the initial conditions and so any errors between the transmitter and receiver wouldgrow exponentially. However, there is a way around this. . .
The transmitter is an electronic circuit where 3 voltages at certain points in the circuit satisfy viaKirchoff’s laws
u = σ(v − u),
v = ru− v − 20uw,
w = 5uv − bw, (68)
77
where σ, r, b are circuit parameters. These equations are just the Lorenz equations (14) under therescalings
u =1
10x, v =
1
10y, w =
1
20z.
At the receiver, the received signal u(t) replaces the “natural” value of the receiver equations
ur = σ(vr − ur),vr = ru(t)− vr − 20u(t)wr,
wr = 5u(t)vr − bwr, (69)
where we have written u(t) to emphasise that the receiver circuit is being driven by the chaoticsignal coming from the transmitter.
Parameters in both (68) and (69) are the same and are chosen such that (u, v, w) behave chaotically.The amazing result is that the receiver circuit quickly synchronises perfectly with the transmitter,starting from any initial conditions!
To be clear, if we let t = (u, v, w) and r = (ur, vr, wr), then the error between the transmitter andreceiver
e = t− r,
tends to zero i.e., e→ 0 as t→∞. This happens despite the receiver only knowing part of t (turnsout synchronisation doesn’t work for all parts of d. . . )
To show this synchronisation we need something called a Lyapunov function of e
Definition 23 (Lyapunov function) Let
x = f(x),
be a dynamical system with an equilibrium at x∗.
A Lyapunov function L(x) is a continuously differentiable, real valued function, with the followingproperties
1. L(x) > 0 for all x 6= x∗, and L(x∗) = 0.
2. L(x) < 0 for all x 6= x∗ (i.e., all trajectories flow “downhill” towards x∗).
Then x→ x∗ as t→∞ and x∗ is said to be globally asymptotically stable.
Intuitively, L(x) looks like a bowl, which all trajectories flow down towards x∗. Note that x∗ is
L(x)
x(t)
x∗
globally stable not just linearly stable. This is the “magic” formula needed to show that the errore between the transmitter and the receiver tends to zero!
Problem: How do we find a Lyapunov function?
The is no systematic method for finding Lyapunov functions (usually divine inspiration, guess worketc.). Thankfully, one does exist for our synchronisation problem. . .
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To find a Lyapunov function L(e), we proceed as follows. Subtract (68) from (69) to get e = t− r
e1 = σ(e2 − e1), (70)
e2 = −e2 − 20u(t)e3, (71)
e3 = 5u(t)e2 − be3, (72)
where e1 = u− ur etc. So u(t) is the chaotic signal, we need to get rid of it by doing the following:take (71)×e2 + (72)×4e3 to get
e2e2 + 4e3e3 = −e22 − 4be23. (73)
Now taking the Lyapunov function to be (guess)
L(e) =1
2
(1
σe21 + e22 + 4e23
). (74)
Observe that L(e) > 0 for all e 6= 0 (due to all the square terms) and L(e) = 0 if e = e∗ = 0.Furthermore, we have
L =1
σe1e1 + [e2e2 + 4e3e3],
= −[e1 −1
2e2]− 3
4e22 − 4be23, using equations (70) and (73),
≤ 0,
provided b ≥ 0. Hence, L(e) in (74) is a Lyapunov function, so e→ 0 as t→ 0 implying r→ t (thereceiver synchronises with the transmitter).
In fact it does this synchronisation exponentially fast! This is important as rapid synchronisation isnecessary for the desired application of transmitting messages.
Summary:
We have shown that the receiver circuit will synchronise with the transmitter circuit if the drivesignal is u(t). However, for the signal-masking application the drive signal is actually u(t) + m(t)where m(t) is the message and u(t)� m(t) is the mask. There exists no proof that the receiver willregenerate u(t) precisely (due to the mask and noise). Hence, why the received message sounds alittle fuzzy.
To test out this idea of sending messages using chaotic synchronisation, we choose the message tobe a sine wave m(t) = A sin(ωt). The figure below shows the recovery of the sine wave message forA = 0.0001, ω = 1. We see the error of the recovered message decays e ∝ e−1.8t and levels out atroughly 0.0003.
0 1 2 3 4 5 6 7 8 9 10- 25
- 20
- 15
- 10
- 5
0
5
recovered messageoriginal message
m(t) = 0.0001sin(t)
m(t) = 0.0001sin(t)log(error)
tt0 2 4 6 8 10 12 14 16 18 20
- 16
- 14
- 12
- 10
- 8
- 6
- 4
- 2
0
2
4
y ≈ −1.8057t + 1.863
Figure 33: Recovery of sine wave message. Matlab codes: [mask-lorenz.m] and [Masking].
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11 Matlab codes
Matlab will be used throughout this course and will build on the courses [MAT1025E] (Experiment)and Numerical & Computational methods [MAT2001].
For a “crash course” in Matlab the [Intro] pdf will help. Several other Matlab introductions can befound here:
[http://personal.maths.surrey.ac.uk/st/D.J.Lloyd/MAT3007.php]
The following PDFs will provide help on
[Iterated Maps] (Covering Cobwebs, Logistic Map and Newton’s method)
[Differential Equations 1] (Covering first-order differential equations)
[Differential Equations 2] (Covering phase planes)
Matlab codes for the following Figures can be found on the Nonlinear Dynamics & Chaos website:
Figure 3 [Chaotic Rabbits] Figure 7 [Sensitive dependence]Figure 8 [Periodic orbit of Logistic map] Figure 12 [pplane7.m]Figure 16 [Brusselator] Figure 17 [pplane7.m]Figure 18 [pplane7.m] Figure 19 [Transient Chaos]Figure 20 [Rossler simulations] Figure 22 [Poincare map of Rossler system]Figure 23 [Rossler system 1D map] Figure 27 [Bifurcation diagram: Logistic map]Figure 28 [Intermittency] Figure 29 [Cobweb Intermittency]Figure 30 [Lyapunov exponent Logistic map] Figure 31(a) [Bifurcation diagram: Rossler]Figure 31(b) [Bifurcation diagram: Sine map] Figure 32 [Koch snowflake]Figure 33 [Masking]
You will also need to download the following to run some of the above codes
Lorenz system [lorenz.m] Brusslator system [brusselator.m]Rossler system [rossler.m] Poincare map [rossler-events-poincare.m]Lorenz map [rossler-events.-lorenz.m] Koch step [kochstep.m]Masking system [mask-lorenz.m] Cobweb [cobweb.m]
Figures 27,31 were actually created using the following matlab codes and imported into iphoto toenhance the image:
Bifurcation diagram: Logistic map [Logistic-diagram.m]Bifurcation diagram: Rossler system [Rossler-diagram.m]Bifurcation diagram: Sine map [Sine-diagram.m]
Warning: These codes take a very long time to finish!
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To run these files, go to the Chaos & Fractals website
[http://personal.maths.surrey.ac.uk/st/D.J.Lloyd/MAT3007.php]
and right-click on the files to save them all in a directory (say Figures)
Now, load up Matlab and change Matlab’s current directory to Figures by clicking on the buttoncircled below. Now just type the name of the Figure you wish to re-create!
References
[1] S. J. Hogan, Lecture notes for advanced nonlinear dynamics and chaos course. University ofBristol, Department of Engineering Mathematics, 2002.
[2] B. Krauskopf and H. Onsinga, Lecture notes for nonlinear dynamics and chaos course.University of Bristol, Department of Engineering Mathematics, 2009.
[3] J. D. Murray, Mathematical biology. I, vol. 17 of Interdisciplinary Applied Mathematics,Springer-Verlag, New York, third ed., 2002. An introduction.
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[4] I. Stewart, Does God play dice?, Basil Blackwell, Oxford, 1989. The mathematics of chaos.
[5] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chem-istry, and Engineering, Westview, 2000.
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