The twist coefficient of periodic solutions of a time-dependent Newton's equation

Journal of Dynamics and Differential Equations, Vol. 4, No. 4, 1992

The Twist Coefficient of Periodic Solutions of a Time-Dependent Newton's Equation

Rafael Ortega I

This paper gives sufficient conditions for the existence of periodic solutions of twist type of a time-dependent differential equation of the second order. The concept of periodic solution of twist type is defined in terms of the corresponding Birkhoff normal form and, in particular, implies that the solution is Lyapunov stable. Some applications to nonlocal problems are given.

KEY WORDS: Periodic solutions; normal forms; twist theorem; Lyapunov stability. AMS SUBJECT CLASSIFICATIONS: Primary 34C25, 34D20o

1. INTRODUCTION

Consider the scalar equation

x" + f(t , x )=0 (*)

where f is T-periodic in t and sufficiently smooth. Let go be a T-periodic solution. It is well-known that the stability

properties of go are not determined by the linear variational equation. In fact the behavior of the solutions of (*) in a neighborhood of go will depend on the nonlinear terms of the Taylor expansion o f f The theory of normal forms combined with the Moser twist theorem provides a method of proving that go is Lyapunov stable in certain cases (Siegel and Moser, 1971, p. 226). We shall say that go is of twist type if the twist coefficient at go of the Poincar6 operator associated to ( , ) is not zero. This coefficient corresponds to the first nonlinear term in the Birkhoff normal form of an area-preserving map.

l Departamento de Matemfitica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada. Spain.

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1040-7294/92/10004)651506.50/0 �9 1992 Plenum Publishing Corporation

652 Ortega

In this paper we prove the existence of periodic solutions of twist type under assumptions that depend on the derivatives of f up to the third order and, especially, on the signs of these derivatives. From the point of view of stability theory, these results can be seen as examples of "a method of the third approximation." It is important to remark that the assumptions are not of the kind of small parameters and that they do not imply that ( . ) is close to an autonomous equation. It follows from the Birkhoff fixed-point theorem and from KAM theory that, around a solution of twist type, there exist subharmonic solutions with minimal periods going to infinity as well as many quasi-periodic solutions (see Siegel and Moser, 1971; Arnold and Avez, 1968). Therefore some consequences on the existence of solutions of these classes can be derived from this paper.

These results are applied to a pendulum of variable length and to a periodic problem of Ambrosetti-Prodi type. The existence of solutions of the periodic Ambrosetti-Prodi problem was analyzed by Fabry etal. (1986). The stability properties of these solutions were studied by Ortega (1989, 1990), assuming that ( . ) was perturbed by a linear friction. A partial extension to the nonfriction case of some of the results is obtained.

The paper is organized as follows. In Section 2 we recall some facts on normal forms of area-preserving maps which are needed and state the main results. In Section 3 we combine the results of Section 2 with some of the techniques developed by Ortega (1989) to obtain the applications. Section 4 is divided into four parts and devoted to the proofs of the results in Section 2. Section 4.1 is concerned with the computation of the twist coefficient of an abstract area-preserving map. In Section 4.2 the derivatives of the third order of the Poincar6 map are found. Section 4.3 shows a useful property of the monodromy matrices of Hill's equation. Finally, the proofs are completed in Section 4.4.

Throughout the paper we use the following notation:

Ixl = Euclidean norm of a column vector x ~ R 2

R[O] = (cos 0 - s i n 0~, \ sin 0 cos 0 /

rotation of angle 0 e

Given g, h: g2 ~ R, I2 measure space, g <~ h r g(x) <~ h(x), Vx E g2, g ~ h r g ~< h with strict inequality on a set of positive measure. 8i is the partial derivative with respect to x;, O0.=O;~j. C~ g~), m>>. 1, class of continuous functions f = f ( t , x ) which are T-periodic in t and have continuous derivatives with respect to x of the orders not exceeding m. an=zr/nT, n = 1, 2, 3,....

The Twist Coefficient of Periodic Solutions of a Newton's Equation 653

2. NORMAL FORMS AND TWIST PERIODIC SOLUTIONS

From now on assume that f ~ c~ x ~), m >~ 4, and let q~ be a T-periodic solution of (.). The corresponding variational equation is given by

y" +fx(t, q~(t)) y = 0 (2.1)

and has Floquet multipliers )~, 22, satisfying 2~22 = 1. It is said that q~ is hyperbolic if the multipliers satisfy I)~] < I < ]22]

and ~p is elliptic if ]2A=l , 2ir i = 1 , 2 . Also, ~p will be called n-elementary if 2 p :~ 1, 1 ~< p ~< n, i = 1, 2.

The method of the linear approximation implies in this case that a hyperbolic solution is unstable, but elliptic periodic solutions are not always stable in the sense of Lyapunov. Therefore, to study the stability, one has to consider the nonlinear terms of the Taylor expansion. The theory of normal forms reduces these terms to a simpler form. We shall collect some facts on area-preserving maps of the plane with positive orientation, also called symplectic, and their normal forms.

Let F = F ( ~ ) be an area-preserving map from a neighborhood of ~0 ~ ~2 into ~2 of class C41 Assume that ~o is a fixed point of F and that the Jacobian matrix F'(~o) is conjugate in the symplectic group to RIO] for some 0~(0 ,2~] . If 0~2kn/n , n=1,2,3 ,4 , l<<.k<~n, it follows from Birkhoff's results that there exist/~ ~ ~ and a symplectic map tp of class C ~ with ~(~o)= ~o such that

-[-O(]ff--~0] 4) as ~--*~o (2.2)

(see Siegel and Moser, 1971, p. 173; Arrowsmith and Place, 1990, p. 305). Moreover,/~ is a symplectic invariant in the sense that it is independent of the choice of the symplectic change of variables ~6 This fact is proved by Markus and Meyer (1980). We use the notation/~ =/~(F, ~o).

Given ~ ~ B 2, let x(t, ~) be the solution of (*) satisfying (x(0), x ' (0)) t= ~. The Poincar6 operator, defined on some open subset of ~2, is given by

P( ~) = (x( T, ~), x'( T, ~) )'

It is well-known that P is area-preserving and that the initial condition of ~p at t = 0, fro = (~p(0), ~p'(0))', is a fixed point of P. In addition, P is of class C m and the eigenvalues of P'(~o) are the Floquet multipliers of (2.1). Now, if ~p is elliptic and 4-elementary, we are in a position to apply the normal form given by (2.2) to the case F = P.

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Definition 1. Let ~0 be a T-periodic solution of (*). We say that q~ is of twist type if it is elliptic and 4-elementary and/~(P, ~o) r 0.

It follows from Moser twist theorem that when f is sufficiently smooth (m large enough), every periodic solution of twist type is stable in the sense of Lyapunov. A discussion on the regularity required in the twist theorem is given by Arrowsmith and Place (1990, p. 332).

In contrast to the definition of elliptic type, the concept of solution of twist type is going to depend on the derivatives of f up to the third order.

We now assume that f ( t , 0) = 0, - ~ < t < ~ , and write f i n the more convenient form

f ( t , x ) = a ( t ) x + b ( t ) x 2 + c ( t ) x 3 + r ( t , x ) , ( t , x ) E ~ 2 (2.3)

where a, b, e~ C(~/TY_) and r(t, x ) = O(Ixl 4) as x ~ 0, uniformly in t E ~. In the next results we prove that the solution x - 0 is of twist type

under certain assumptions on a, b, c. They do not impose restrictions on the size of b and e.

Theorem 1. Assume that f is given by (2.3) and x =- 0 is elliptic as a T-periodic solution of (*). In addition,

2 (i) a ..~ a3,

(ii) b<~Oorb>~O,

(iii) c <~ O, T (iv) ~o[Ib[+lcl]>O.

Then x =-0 is of twist type.

The next result shows that the previous theorem can be sharpened if f has an odd expansion.

Theorem 2. Assume that f is given by (2.3) with b =- 0 and let x =- 0 be elliptic. In addition,

(i) a ~ a 2,

(ii) c ~O or c>>O.

Then x - 0 is of twist type.

Remarks 1. 1. Assumption (i) in Theorem 2 is natural since the solution x = 0 is not 4-elementary for a=~rz z. We do not know if the constant a32 appearing in (i) of Theorem 1 can be improved.

2. Conditions (ii), (iii), and (iv) in Theorem 1 are not sufficient to conclude that every elliptic solution is stable. The next result shows, in

T h e Twist Coefficient of Periodic Solutions o f a Newton's Equation 6 5 5

particular, that some restrictions on the size of a(t) must be imposed to get stability.

An Example of Instability. Consider

x" + (2zt/3 T) 2 x + b(t) x 2 + s(t, x) = 0

with b t C(~/T7/), s t C o, 3 ( N/TZ x N), s(t, x) = O (1 x[ 3 ) as x ~ 0, uniformly in t t N. Then x - 0 is unstable if

Io bhl + I f bh2 > 0 (2.4)

where h I = --~t 3 + 3~b~b2, h2 = -~k~ + 3~k~kl, with ~/l(t) = cos( 2~zt/3 T), ~b 2( t ) = sin( 2nt/3 T).

The proofs of the results of this section are given in Section 4.

3. APPLICATIONS

3.1. The Periodic Ambrosetti-Prodi Problem

In this section we consider the equation

x" + g(x) = p(t) + s

where g t C4(~), p t C(~/TY_), s t ~. Assume that g is coercive, that is,

lim g ( x ) = + ~

(3.1)

(3.2)

It was proved by Fabry et al. (1986) that there exists Sot E (depending on g and p) such that (3.1) has

at least two T-periodic solutions if s > So,

at least one T-periodic solution if s = So, and

no T-periodic solution if s < So.

Massera's theorem and a truncation argument given by Ortega (1989) prove, in addition, that every solution of (3.1) is unbounded if s < So.

This result is inspired by the paper of Ambrosetti and Prodi (1972). The original result was obtained for a Dirichlet problem and replaced the coercivity condition (3.2) by more restrictive assumptions on convexity and

656 Ortega

jumping of eigenvalues. These conditions can be translated to the periodic case as

g"(x)>O, V x e ~ (3.3)

- ~ ~< g'( - oo ) < 0 < g'( + co ) ~< (2~/T) 2 (3.4)

[Note that 0 and (2n/T) 2 are the first two eigenvalues of the operator - d 2 / d t 2 acting on T-periodic functions.]

On the other hand, if (3.3) and (3.4) are assumed, the conclusion can be sharpened and (3.1) now has exactly two T-periodic solutions if s >So and exactly one if s = So. A proof can be obtained adapting the arguments of Ortega (1989, 1990). In the process one obtains some additional information on these periodic solutions. In fact, assuming that (3.3) and (3.4) hold and S>So, the two T-periodic solutions ~ol and q~2 are 1-elementary and the corresponding indexes satisfy

~)T(q)i)=(--1) i, i = 1 , 2

[Here 7T refers to the index of a T-periodic solution as given by Krasnoselskii (1968)]. Additional information is now obtained when inequalities more restrictive than (3.4) are assumed.

Proposition I. Assume that g satisfies (3.3) and

- ~ ~< g ' ( - ~ ) < 0 < g ' (+ ~ ) ~< a~ (3.5)

Then, for each S > So, ~ol is hyperbolic and ~o z is elliptic.

The proof follows from the previous remarks together with the result below, which concerns Eq. (,).

Lemma 1. Let q~ be a 1-elementary T-periodic solution of ( , ) and assume that

L( . , )

Then q~ is elliptic (resp. hyperbolic) i f and only tfTr(q~)= 1 (resp. - 1 ) .

This lemma is inspired by Theorem 1.1 of Ortega (1989) and the proof is essentially the same.

In order to apply Theorem 1 in combination with Proposition 1, we impose

g"(x) > 0 and g"(x)<.O V x e ~ (3.6)

--oo ~< g'(-- oo) < 0 < g ' (+ oo) ~< a~ (3.7)


Theorem 3. Assume that (3.6), (3.7) hold and s > so. Then q~2 is o f twist type.

A model equation for the application of these results is

x " + , ~ x + e X = p ( t ) + s , 2 > 0

For ~. ~ (0, a~] and s > So, it follows from Theorem 3 that ~P2 is stable. Also, the Birkhoff fixed-point theorem implies the existence of infinitely many subharmonics with periods going to infinity. Note that for s <so subharmonic solutions cannot exist since every solution is unbounded. Fonda etal . (1989) used a variational method to prove that there exists sl (possibly greater than so) such that there exist infinitely many subharmonics for s > sl when 2 ~ (0, (2r~/T)2].

For 2~ (a~, (2re/T) 2] it is possible to adapt some techniques from Ortega (1990) and prove that for some p and s > s o , q~2 is hyperbolic.

For 2 ~ (a 2, a 2] it is an open question to decide if ~P2 is always stable.

3.2. A Pendulum with Variable Length

Consider the equation

x" + ~(t) sin x = 0 (3.8)

with ~ e C(~/T~_), ~ >~ O. This equation models the motion of a pendulum with variable length, also called a swing. Arnold and Avez (1968, p. 85) proved that if ~ is sufficiently close to a certain constant, then x - 0 is stable. We obtain some properties of the equilibrium positions x = 0 and x --- rc under certain restrictions on ~ of a different kind. First we linearize.

Proposition 2. Assume that 0 ~ c~ ~ a21 . Then x =- 0 & elliptic and x =- rt is hyperbolic.

Proof. The linear variational equation is

y" + ~ct(t) y = 0 (3.9)

w i t h e = l i f x - 0 a n d e = - I i f x = r t . For 8 = 1 it follows from classical results (see Magnus and Winkler,

1979, p. 68) that (3.9) is stable. Since 0 ~ e every periodic solution should oscillate, but Sturm comparison theory implies that the distance between two consecutive zeros is greater than T. In consequence, (3.9) cannot have T-periodic solutions and 1 is not a multiplier. A similar reasoning proves that (3.9) does not have skew-periodic solutions and - 1 is not a multiplier. This implies that x - 0 is elliptic.

658 Ortega

For e = - 1 the result follows directly f rom Cesari (1971, p. 60). We now use the expansion ~(t) sin x = ~(t) x - ~(t) x3/6 + . . . and

apply Theo rem 2 to obtain.

Theorem 4. Assume that 0 ~ ct ~ a~. Then x =- 0 is o f twist type.

Remark 2. We do not know which is the largest constant c such that x - 0 is stable if 0 ~ ~ ~ c. Of course c >~ ~2.

4. PROOFS

4.1. A Formula for the Computation of the Twist Coefficient

In this section we consider maps F ~ C4(~'~, ~2), F = (F~, F2) t, where g2 is some open ne ighborhood of 0 in ~2 and assume that

det F ' (~) = 1, u ~ s and F(0) = 0

We use the no ta t ion F * = F ' ( 0 ) l o F and derive a formula for the c o m p u t a t i o n of fl(F, 0) in terms of the real derivatives of F and F*.

Proposition 3. Assume that

F'(O) = R[O] with 0 ~ (0, 2~z],

0 v ~ 2k~z/n,

Then

n = 1 , 2 , 3 , 4 ,

fl(F, 0) = (1/16) c u r l ( A F * ) - - 3 sin 0(1 - - cos 0) 1

x J s + ( F ) 1 2 - s i n 30(1 - c o s 30) -1 ]s (F)l 2

where all the derivatives are evaluated at 0 and

l ~ k ~ n (4.1)

(4.2)

8s+ (F) = ((tgal - 022) F1 _+ 20a2F2, - [(022 - t~la) F2 _+ 2012F, 3) '

[We also recall that, given a vector field A = ( A I , A 2 ) t, c u r l ( A ) = 0 1 A 2 - O 2 A 1 . ]

Proof. We int roduce the complex variables z = x + iy, f = x - iy, and expand F in the form

F(z, f ) = 2z + F2(z, f ) + F3(z, f ) + O(IzP 4)

with 2 = e ~~ F2(z, 5) = a z 2 -~- bz f + cf 2, F3(a, f ) = mz 3 + nT~Zz ~- p z z 2 Ji- q~3

and a, b, c, m ..... complex constants.


The normal form of N is expressed in complex notation as

N(z, 2) = 2z exp(i//Izl 2) = 2 z + i2flz22 + O(Izl 4)

Since F'(0) is a rotation, the change of variables ~ can be chosen in such a way that ~u'(0)= I (see Siegel and Moser, 1971) and expanded as

~'(z, 2 ) : z + ~V2(z, 2) + ~3(z, 2) + O(Izl 4)

The identity (2.2) is equivalent to

F(~, ~')= ~(N, N) + O(Izl 4)

and, equating terms, it implies

2 ~ 2 ( Z , 2) - - ~2(2Z, 2Z) + F2(z, 2) = 0 (4.3)

2~3(z, 2) - ~3(2z, 2z) + 2az~2(z, 2) + b(z~2(z, 2) + 2~2(z, 2))

+ 2c27t2(z, 2) + F 3 ( z , 2) : i2fiz22 (4.4)

From (4.3),

~2(z, 2 ) : ( 2 2 - 2 ) l a z 2 + ( 1 _ 2 ) - l b z 2 + ( i 2 - 2 ) - lc22

Identifying the coefficient of z22 in (4.4), one obtains

i 2 f l = n + 2 [ c l Z ( 2 2 - 2 ) - l + [ b [ 2 ( 1 - 2 ) - l + a b ( 1 - 2 2 ) ( 2 2 - 2 ) 1 (4.5)

Since F is area-preserving the coefficients of F2 are related by the identity

22a + 2b = 0 (4.6)

Combining (4.5) and (4.6), ifl =)In + 2 ]c[ 2 ( 2 3 - 1) -1 + 2 [ai 2 (1 + 22) ( 2 - 1)-1. The constant fl is real and this fact together with a computation shows that

f l= Im()~n)- 3 sin 0(1 - c o s 0) 1 [al2 sin 30(1 - c o s 30) 1 [C[2

Now, passing from complex to real derivatives, a=OzzF(O)/2=s+(F), c=O~F(O)/2=-s (F), ~n=Oz~e2F(O)/2=(1/16)[div(AF*)+icurl(AF*)] at ~ = 0, and the previous formulae lead to (4.2).

Remark 3. A similar computation was performed by Ioos (1979, p. 30) and Wan (1978) in the context of Hopf bifurcation. They considered arbitrary two-dimensional maps, so that (4.6) does not necessarily hold, and arrived at a formula similar to (4.5). A formula for the twist coefficient

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of symplectic maps was obtained by Moeckel (1990). However, it was assumed that, in addition to (4.6), the 2-jet of F satisfied l a l = Ic l , which is not true for generic area-preserving maps.

We conclude this subsection with two consequences of the previous formula that are used in the proofs of the results of Section 2. In the rest of the paper the inequality x > 0 for a vector x = (x~, X2) t m e a n s xi > 0, i = 1,2.


F ' ( O ) = D , R [ - O ] D~ -1, 0 ~ (0, ~/3 ]

Here ~ > 0 and D~ = diag(~, ~ - ' ). In addition,

J0akF*(0) > 0 for each triplet 1 <~ i, j, k <~ 2

Then fl(F, O) > O.

Proof. If F ' (0) = R [ - 0 ] , the result is a consequence of (4.2). In the general case we define G = D ~ o F o D ~ . Then G' (O)= R[ -O] , and since D~ is symplectic, fl(F,O)=fl(G,O). It is easy to verify that G * = D ~ l o F* o D, , implying that JOokG*(O)> 0. We now apply the previous discussion to G.


F ' ( O ) = D ~ R [ - O ] D2 ~, 0 ~ (0, ~/2)

In addition, O,~F(0)=0 for each couple 1~<i, j~<2, and JOokF*(O)>O (resp. < 0 ) for each triplet 1 <<. i, L k <. 2. Then fl(F, O) > 0 (resp. <0) .

The proof is similar. It must be noted that, in this case, s+_(F)= O.

4.2. The 3-Jet of the Poincar~ Map

In this section we assume that ~p - 0 is a solution of (*) a n d f i s given by (2.3). We compute the derivatives of P up to the third order at the fixed point ~ = 0.

First we introduce some additional notation. Let ~bt,~b2 be the solutions of the linear equation

y" + a(t) y = 0 (4.7)

satisfying ~bl(0) = ~b[(0) = 1, ~b[(0) = 42(0) = 0.


The matrix

~, . /01(t) O2(t)'~

is symplectic and such that P ' (0 )= q~(T). To compute the derivatives of higher order we consider the non-

homogeneous initial-value problem

y" + a(t) y + p(t) = 0, y(0) = y'(0) = 0 (4.8)

Given 1~<i, j~<2, let ~bij(t) be the solutions of (4.8) for P=-Pu with p,j = 2b~b~bj. Then ~ P ( 0 ) = (q~o.(T), ~.(T))'.

In the same way, given a triplet t ~< i, j, k ~< 2, let ~bijk(t) be the solution of (4.8) for p-pij~ with pijk=2b[~)ijO~+~k~j+q)kj~]+6CqJiOj(~k. Then ~3~kP(O) = (~k(T), ~k(T))'.

Applying the formula of variation of constants, we obtain the expansion

P(~) = qb(T)[~ + ~,~jJd~/2! + ~i~j~kJdUk/3!] + O(l~l 4)

where the summation convention is used and

diJ= fTpij(s)(()l(s), ~)2(s))t ds, diJk= forp~jk(S)(~,(s), O:(s))t ds

We conclude this subsection with a result on the signs of d ~jk.


a <~ a~ (4.9)

and (ii), (iii), and (iv) of Theorem 1 hold. Then d ijk < 0 for each triplet l <~i,j,k<~2.

The proof follows after two lemmas.

Lemma 2. Assume that (4.9) holds. Then (~ > 0 on (0, T), i = 1, 2.

Proof. For each r > 0 the inequality below holds:

f~ Y'(s)2 ds>~(Tt/2~)2 f~ y(s)2 ds,

VyeHl(0, 'c) , y ( 0 ) = 0 or y ( r ) = 0 (4.10)

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This follows from the variational characterization of the first eigenvalue of the Sturm-Liouville problems y" + 2y = 0, y(0) = y'(r) = 0 [or y'(0) = y(~)=0].

By contradiction, assume that for some r s (0, T), bg(z) = 0. Multiplying (4.7) by ~ and integrating over (0, r),

Io (b~(s)2 ds= fo a(s) q~i(s) 2 ds ~ a~ fo

a contradiction with (4.10) since a2 < n/2r.

~i(s) 2 ds

Lemma 3. Assume that (4.9) holds and y is the solution of (4.8) with p~C[O, T], p<~O. Then y>>.O on [0, T]. Moreover, if p(to)<O for some t o ~ (0, T), then y(t) > 0 for each t ~ [-t o, T].

Proof. The function y is given by y(t)= -Sto G(t, s)p(s)ds, where G(t,s)=r O~s<<,t~T. Since ~bl>0 on (0, T) by the previous lemma and (~bg/~bl)'=det~/~bl 2= 1/~b~>0, it follows that G(t,s)>O if 0 < s < t < 7".

Proof of Proposition 6. Lemma 2 implies that sign(p,7 ) = sign(b). By Lemma 3, b~b, 7 <~ 0 if b ~ 0, and again, Lemma 2 leads to the conclusion.

4.3. A Monodromy Matrix of Hill's Equation

We consider the linear equation (4.7) and assume that the Floquet multipliers are 2 and 2 with 2 = e i~ 0 > 0, 0 r nn, n = 0, 1, 2 ..... For each toe ~, let ~(t, to) denote the matrix solution of Y ' = A ( t ) Y , Y(to)=I, where A(t) = (_~ m ~). The matrix ~(to + T, to) is a monodromy matrix of (4.7) and belongs to the class of matrices which are similar to R[ _+ 0]. The following result shows that r + T, to) can be chosen in a smaller class.

Proposition 7. In the previous assumptions there exist to and ~ > 0 such that either

(i) q~(to+ T, to)=D~R[O] D21 or (ii) q~(to+ T, t o ) = D , R [ - O ] D~'

with D~ = diag(~, a - l ) .

Remark 4. The first alternative (i) occurs when 2 is Krein-negative and (ii) occurs when 2 is Krein-positive, in the sense of the theory of linear Hamiltonian systems (see Ekeland, 1990).

In the proof of the proposition we use the following elementary result.


Lemma 4. Let M be a real 2 • 2 matrix with eigenvalues 2 and ~ and eigenvector w e C 2 - {0} such that Mw=2w. Define P = ( R e w [ - I m w). Then M = PR[O] P-~.

Proof. Let ~b(t) be a nontrivial (complex-valued) solution of (4.7) such that ~b(t + T) = 2qt(t), t ~ ~. Since t ~ L~b(t)[ 2 is real-valued and T-periodic, there exists to~N such that (d/dt)I~b(t)12=0 at t=to. Also, ~b(to) ~0. [-Otherwise (~(t)/O'(to)= )~(t) is a real-valued solution satisfying ;~(t+ T ) = 2Z(t), contradicting 2 r ~] . Define O ( t ) = O(t)/(~(to). Then O( to )= l and (d/dt) lO(t)l 2 = 0 at t = t o . Setting ~ = ~ 1 + i ~ 2 , one has Ol ( to )= l , O2(to)=0, O](to)=0. Now (O(to),O'(to)) t is an eigenvector of qb(to+T, to) and Lemma4 implies that q)(to+T, t o ) = P R [ O l P -1 with P = d i a g ( 1 , - ~ ; ( t o ) ) . If O~(to)<0, alternative (i) holds with e = [O;(to)[-m. If O;(to)>0, the whole reasoning is repeated replacing 2 by J. and ~b by q~, and the second alternative holds.

2 Proposition 8. In the previous setting assume, in addition, that a <~ a n, n >1 2. Then there exists to and ~ > 0 such that q~(to+ T, to )= D~R[-O] O~ ~ with 0 ~ (0, n/n].

2 First, we Proof. The case a = a] is trivial so that we assume a,~ a n. prove that the multipliers of (4.7) satisfy

2 = e i~ [0[ <n/n (4.11)

Consider the parametric equation y" + ~a(t) y = 0, e ~ [0, 1 ], with discriminant zl[e] (cf. Magnus and Winkler, 1979). To prove (4.11) it is sufficient to verify the inequality A[,1] > 2 c o s n/n. Assume first that A[e] = 2 cos n/n for some ~ [-0, 1]. Then there exists a nontrivial solution ~b of the parametric equation such that ~b(t + T) = ei'/"~k(t), t ~ ~. In consequence, ~b(t + nT)= -~k(t), implying that ~ is skew periodic (period nT). Sturm comparison theory applied to the real and imaginary parts of ~, shows that this is not possible if a ~ a2; see Lemma 1.4 of Ortega (1989). In consequence, A [ e ] ~ 2 c o s n / n for each e~[0 , 1]. It is known that depends continuously on ~, and since A[ ,0]=2, the proof of (4.11) is finished.

We now apply Proposition 7 and obtain to and ~ > 0 such that �9 (to+ T, to)=D~R[ +_0] D~ 1, O~ (0, n/n]. Lemma 2 implies that the coefficients of the first row of ~b(to + T, to) are nonnegative so that the rotation R must be of angle - 0 .

664 Ortega

4.4. Conclusion of the Proofs

Proof of Theorem 1. Let t o ~ ~ be given as in Proposition 8 (n = 3) and make the change of the independent variable z = t - t o . Then ( , ) is transformed into

(d2/& 2) x + g(z, x) = 0 (4.12)

where g(y, x) = f ( t 0 + r, x) = a(to + ~) x + b(to + ~) x 2 + C(to + r) x 3 + .... Denote by Q the Poincar6 operator associated to (4.12). We have

Q ' ( O ) = D , R [ - O ] D~ 1, 0e (0 , z/3]. Now,. according to Section4.2, OokQ*(O)=Jd ~, and since the coefficients of g are translates of a, b, and c, Proposi t ion6 is applicable and d~ Thus JaokQ*(O)=J2d ijk= - d U~ > 0 and, from Proposition 4,/?(Q, 0) > 0.

Now Q and P, the Poincar6 map of (.) , are symplectically conjugate, implying/~(Q, 0) =/~(P, 0), and the proof is finished.

The proof of Theorem 2 is similar, Proposition 5 is used.

Proof of the Example of Instability. We use the following instability criterion, due to Levi-Civitfi (Siegel and Moser, 1971, p. 222): let F e C3(0, R 2) be such that F ( 0 ) = 0, where f2 is some open neighborhood of 0 in N 2. Assume that, in complex notation,

F(z ,~ )=r 3) as z ~ 0

where o92 + ~o + i = 0 and c r 0. Then z = 0 is unstable as a fixed point of F. The translation of these assumptions into real notation is

f'(O)=R[+__2rc/3], (011-022)f-2JO12F=/=O

We apply this result with F = P and assume that T = 2r~/3. (This is not restrictive, thanks to the change of the time-variable v=27rt/3T). Then P'(O) = R [ - 2 ~ / 3 ] and, since it commutes with J,

(011 - - 022) P - 2JOI2P = R[ --2rc/3 ](J(d H - d 22) + 2d~2),

which is different from zero if and only if (2.4) holds.

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The twist coefficient of periodic solutions of a time-dependent Newton's equation

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