STABILITY OF SUBHARMONIC SOLUTIONS OF FIRST-ORDER HAMILTONIAN SYSTEMS WITH ANISOTROPIC GROWTH?

Chungen LIU

School of Mathematics and Information Science,Guangzhou University,Guangzhou 510006,China

E-mail:liucg@nankai.edu.cn

Xiaofei ZHANG

School of Mathematics,Nankai University,Tianjin 300071,China

E-mail:835858094@qq.com

Abstract Using the dual Morse index theory,we study the stability of subharmonic solutions of fi rst-order autonomous Hamiltonian systems with anisotropic growth,that is,we obtain a sequence of elliptic subharmonic solutions(that is,all its Floquet multipliers lying on the unit circle on the complex plane C).

Key words Hamiltonian system;the dual Morse index;subharmonic solution;stability

1 Introduction

In this paper,we consider the stability of subharmonic solutions of the following autonomous Hamiltonian system

Now we state the main results as follows.

Theorem 1.1Suppose that H satis fi es(H1)–(H6),then for every j ∈ N,system(1.1)possesses a 2jτ-periodic solution z2jwhich are elliptic.

From[15],we see the subharmonic solutions sequence{z2j}has a geometrically distinct in fi nite subsequence.

Using the truncation method as in[2,15]and proceeding as Theorem 1.1(see[10,15]),we have the following two similar conclusions.

Corollary 1.2Suppose that H satis fi es(H1),(H2),(H4)–(H6)and

(H3)′There exist constants ξi, ηi>0 with ξi+ ηi=1(i=1,2, ···,n)such that

then we have the same result as in Theorem 1.1.

Corollary 1.3The conclusion of Theorem 1.1 still holds if H satis fi es

(C1) H∈C2(R2n,R)and H(z)>0 for z 6=0.

(C2) There exist constants 0< θ<1 and R,?i,ψi>0 with ?i+ψi=1(i=1,2,···,n)such that

The above conditions(C1)–(C4)are similar to those of[2]with minor di ff erence.

As[10]points out,tackling the stability of subharmonic solutions of Hamiltonian systems is helpful to studying the global dynamic behaviours in depth,which concerns the Floquet multipliers lying on the unit circle,see[3–6,10–13,16]and the references therein.On the basis of the dual Morse index method in[8]and the index estimate method in[10],we generalize the stability results in[10]in the case where H does not contain the quadratic form.

Our paper proceeds as follows.In Section 2,we present some basic knowledge of Sobolev space and the homological link theorem used in this paper.In Section 3,we prove the existence of a 2τ-periodic solution with Morse index estimate(see[9,14]).In Section 4,we take advantage of this index estimate information to obtain an estimate for the dual Morse index.Finally in Section 5,we introduce the relation between the dual Morse index and the Floquet multipliers from[10],and prove Theorem 1.1.

2 Preliminaries

Next we recall the homologically link theorem in[1].

De fi nition 2.2(see[1]) Let Q be a topologically embedded closed q-dimensional ball on a Hilbert manifold M and let S ? M be a closed subset such that?QTS= ?.We say that?Q and S homologically link if?Q is the support of a non-vanishing homology class in Hq?1(MS).

Recall that the functional f satis fi es the so called Cerami condition((C)condition for short)on J? RS{±∞}if{zm}? M such that f(zm)→ c∈J and(1+kzmk)k?f(zm)k→ 0 as m→+∞has a convergent subsequence.

3 2τ-Periodic Solution with Morse Index Estimate

ProofNote that for z∈Em,odd,we have?fm(z)=Pm?fodd(z)and

where V1(z)is de fi ned in(H2).

Then the proof follows the same procedure as that in[15]. ?

Note that if foddsatis fi es(C)?condition on Eodd,then fmsatis fi es(C)condition on Em,odd.

By Lemma 3.1,we may assume that zm→z∈Eoddwith fodd(z)≥δ and?fodd(z)=0,which implies that f(z)≥ δ and ?f(z)=0.By(H1),we see z is a nonconstant 2τ-periodic solution of system(1.1)and z(t)6=0,t∈R. ?

4 The Dual Morse Index

The statements in this section are similar to those in[8].

Theorem 4.2For the dual Morse index de fi ned above,we have≤ 1.

ProofThe proof follows[8],we omit it.?

5 The Floquet Multipliers and(ω,k,odd)-Index

Consider the following system in the interval[0,τ],that is,?J˙y=ky.Its Floquet multipliers are ω±=exp(±kτi)with multiplicity n respectively,ω+is Krein positive de fi nite and ω?Krein negative de fi nite(see[5]for the de fi nition of Krein de fi nite).

Proposition 5.1(see[10]) The(ω,k,odd)-index jω,k,odd(z)varies only when ω is a Floquet multiplier of γ(τ)or ω±.Moreover if none of the Floquet multipliers equals ω±,then δjω,k,odd(z)=p0? q0,where ω is a Floquet multiplier of γ(τ)with its Krein type(p0,q0)and δjω±,k,odd(z)= ?n.If ω+is also a Floquet multiplier of γ(τ),then δjω±,k,odd(z)=±(p? q?n),where(p,q)is the Krein type of ω+.In addition≤ d,where 2d is the multiplicity of?1 as a Floquet multiplier of γ(τ).

Proposition 5.2(see[10]) jω,k,odd(z)≥ 2n,ω =exp(iθ),θ→ 0.

Proof of Theorem 1.1By Theorem 3.5,z is a 2τ-periodic solution of(1.1).Thus by Theorem 4.2 and Propositions 5.1 and 5.2,if θ→ 0 and ω =exp(iθ),there holds

where n+and n?are total positive and negative multiplicity of the Floquet multipliers lying on the upper unit circle.If ω has the Klein type(p,q),then it has positive multiplicity p and negative multiplicity q.Therefore,we have n?+d≥n?1.Because the Floquet multipliers lying on the unit circle are symmetric and z is degenerate,we obtain 2n Floquet multipliers lying on the unit circle. ?