Boundedness of Solutions for Duffing Equation with Low Regularity in Time?

Xiaoping YUAN

1 Introduction

In 1962,Moser[6]proposed to study the boundedness of all solutions(Lagrange stability)for Duffing equation

where β >0,α ∈ R are constants.

In 1976,Morris[5]proved the boundedness of all solutions for

Subsequently,Morris’boundedness results was,by Dieckerhoff-Zehnder[1]in 1987,extended to a wider class of systems

Then they remarked that:

“It is not clear whether the boundedness phenomenon is related to the smoothness in the t-variable or whether this requirement is a shortcoming of our proof.”

In 1989 and 1992,Liu[3–4]proved the boundedness for

In 1991,Laederich-Levi[2]relaxed the smoothness requirement of Pj(t)(j=0,1,···,2n)for(1.1)to

In his PhD thesis(1995),the present author further relaxed the requirement to C2(see[12–14]).

In the present paper,we will relax the smoothness requirement to H¨older continuity.More exactly,we have the following theorem

Theorem 1.1 For arbitrary given constantfor n+1≤j≤2n and Pj∈L(T1)for 0≤j≤n.Then every solution x(t)of the equation(1.1),

is bounded,i.e.it exists for all t∈R andwhere the constantdepends the initial data

Remark 1.1 In[11],it is proved that there is a continuous periodic function p(t)such that the Duffing equationpossesses an unbounded solution,which shows that the H¨older continuity of the coefficients Pj’s is necessary for the boundedness of solutions.In this sense,the result is almost sharp.

2 Action-Angle Variable

Replacing x by Ax in(1.1),we get

3 Approximation Lemma

First,we cite an approximation lemma(see[9–10]for the detail).We start by recalling some definitions and setting some new notations.Assume that X is a Banach space with the normFirst recall thatfor 0< μ <1 denotes the space of bounded H¨older continuous functionswith the form

Theorem 3.1 (Jackson-Moser-Zehnder)Letfor some ?>0 with finite C?norm over Rn.Let φ be a radical-symmetric,C∞function,having as support the closure of the unit ball centered at the origin,where φ is completely fl at and takes value 1,and let K=bφ be its Fourier transform.For all σ>0 define

Then there exists a constant C ≥ 1 depending only on ? and n such that the following holds:For any σ >0,the function fσ(x)is a real-analytic function from Cnto X such thatdenotes the n-dimensional complex strip of width σ,

By this theorem,for each,j=n+1,n+2,···,2n,and any ε>0,there is a real analytic function1A complex value function f(t)of complex variable t in some domain in C is called real analytic if it is analytic in the domain and is real for real argument t.fromto C such that

Now let us restrict I to some compact intervals,[1,4],say.Let A?1< ε0.

For a sufficiently small ε0>0,letting

where C is a constant2Denote by C a universal constant which may be different in different place.depending on only

4 Symplectic Transformations

We will look for a series of symplectic transformationssuch thatsuch that Moser’s twist theorem works for

5 Proof of Theorem

By using Picard iteration and Gronwall’s inequality and noting(4.36),we get that the time-1 map of(5.1)is of the form

Since(5.1)is Hamiltonian,the map P is symplectic.By Moser’s twist theorem at pp.50–54 of[7](also see[8]),P has an invariant curve Γ in the annulus[2,3]× T1.Since A can be arbitrarily large,it follows that the time-1 map of the original system has an invariant curve ΓAin the annulus[2A+C,3A?C]×T1with C being a constant independent of A.Choosing a sequence A=Ak→∞as k→∞,we have that there are countable many invariant curves ΓAk,clustering at∞.Therefore any solution of the original system is bounded.This completes the proof of theorem.

Remark 5.1 Any solutions starting from the invariant curves ΓAk(k=1,2,···)are quasi-periodic with frequencies(1,ωk)in time t,where(1,ωk)satisfies Diophantine conditions and

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