XING XIU-MEIMA JINGAND JIAO LEI

(1.School of Mathematics and Statistics,Yili Normal University,Yili,Xinjiang,835000)

(2.School of Science,Nanjing University of Science and Technology,Nanjing,210094)

Boundedness in Asymmetric Quasi-periodic Oscillations

XING XIU-MEI1,MA JING1AND JIAO LEI2,*

(1.School of Mathematics and Statistics,Yili Normal University,Yili,Xinjiang,835000)

(2.School of Science,Nanjing University of Science and Technology,Nanjing,210094)

Communicated by Li Yong

In the paper,by applying the method of main integration,we show the boundedness of the quasi-periodic second order dif f erential equationx′′+ax+?bx?+?(x)=p(t),wherea/=bare two positive constants and?(s),p(t)are real analytic functions.Moreover,thep(t)is quasi-periodic coeffi cient,whose frequency vectors are Diophantine.The results we obtained also imply that,under some conditions, the quasi-periodic oscillator has the Lagrange stability.

boundedness,quasi-periodic,KAM theorem

1 Introduction

In 1999,Fabry and Mawhin[1]suggested studying the boundedness of all the solutions for the following dif f erential equation:

whereare two positive constants,p(t)is a periodic function,x+=max{x,0}andx?=max{?x,0}.Dancer[2]studied the periodic and Dirichlet boundary value problems for(1.1).Many authors are also concerned with the Lagrange stability of this equation.The fi rst boundedness result for(1.1)is due to Ortega.In[3],Ortega obtained the stability for

are f i nite.

Firstly,we introduce some basic notations,properties and lemmas which are used in this paper.

Lemma 1.1[7]The following statements are true:

(i)Let f(t),g(t)∈Q(μ).Then g(t+f(t))∈Q(μ).

Then the inverse relation is given by t=α?1τ+h1(τ)and h1(τ)∈Q(μ/α).

Sun[11]considered the map

whereL,M,fandgare quasi-periodic inxwith the frequencyμand real analytic in the complex neighborhood of R×[a,b].Assume thatf(x,y,0)=g(x,y,0)=0 and theμsatisf i es the Diophantine condition:

for all integerk=(k1,k2,···,km)0.Furthermore,suppose that the mapMδhas the intersection property,that is,for every Jordan curvey=φ(x)which is homotopic to the line satisf i esMδ(Γ)Sun obtained the following two theorems.

Theorem 1.1([11],Theorem 2.3)Ifμ1,μ2,···,μmand2π/α are rational independent and

then there exists Δ＞0such that if0＜δ＜Δ,the map Mδhas an invariant curve y=?(x), where ?(x)is real analytic and quasi-periodic with the frequencyμ.

Theorem 1.2([11],Theorem 2.4)are rational dependent.Lsatisf i es

In addition,there is a real analytic function I(x,y)≡I(x+α,y)satisfying

Moreover,suppose that there are two numbersandwith

where

then there exists a Δ＞0such that if0＜δ＜Δ,the map Mδhas an invariant curve which is of the form y=?(x),where ?(x)is real analytic and quasi-periodic with the frequencyμ, and the constant Δ＞0depends on a,b,?a,?b,L,M and I.

2 Boundedness of Quasi-periodic Oscillations

In this section,we f i rst introduce some notations,second state our main theorem on boundedness of asymmetric quasi-periodic oscillations,last give some of the lemmas and transformations which are used in the proof of Theorem 2.1.

Denote byC(t)the solution of

with the initial condition

The derivative ofC(t)is denoted by?S(t).It is easy to verify thatC(t)andS(t)are of 2πωfortandC(t)is second order dif f erential,where

C(t)is even and can be given by

Moreover,

For a given quasi-periodic functionp(t),let

where

Now we can state our main theorem.

Theorem 2.1Suppose that(1.3)and(1.4)hold.Then

(i)are rational independent and

then all the solutions of(1.1)are def i ned for all t∈Rand

(ii)Ifμ1,μ2,···,μmandare rational dependent and

then all the solutions of(1.1)are def i ned for all t∈Rand

Letx′=?y.Then(1.1)is equivalent to the following system:

and the Hamiltonian function is

where

Throughout this paper,we denoteC＞1 universal positive constants not concerning their quantities.

Next we state Lemmas 2.1–2.2 and Lemma 3.2 in the paper.The proof of Lemmas 2.1–2.3 is similar to the proof of Lemmas 3.1–3.2 and Lemma 4.3 in[12].The conclusion of Lemma 3.2 in this paper can be obtained from the properties 3.1 and 3.2 in[11].

Lemma 2.1The unique solution(x(t;t0,x0,y0),x(t;t0,x0,y0))of(2.2)satisfying

exists on the whole t-axis for all(x0,y0)∈R2and all t0∈R.

Under the transformationΨ:R+×S1→R2