Open Access

Lagrangian Stability of a Class of Second-Order Periodic Systems

Boundary Value Problems20112011:845413

DOI: 10.1155/2011/845413

Received: 24 November 2010

Accepted: 5 January 2011

Published: 11 January 2011

Abstract

We study the following second-order differential equation: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq1_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq2_HTML.gif   ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq3_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq5_HTML.gif are positive constants, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq6_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq7_HTML.gif . Under some assumptions on the parities of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq9_HTML.gif , by a small twist theorem of reversible mapping we obtain the existence of quasiperiodic solutions and boundedness of all the solutions.

1. Introduction and Main Result

In the early 1960s, Littlewood [1] asked whether or not the solutions of the Duffing-type equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ1_HTML.gif
(1.1)

are bounded for all time, that is, whether there are resonances that might cause the amplitude of the oscillations to increase without bound.

The first positive result of boundedness of solutions in the superlinear case (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq10_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq11_HTML.gif ) was due to Morris [2]. By means of KAM theorem, Morris proved that every solution of the differential equation (1.1) is bounded if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq12_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq13_HTML.gif is piecewise continuous and periodic. This result relies on the fact that the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq14_HTML.gif can guarantee the twist condition of KAM theorem. Later, several authors (see [35]) improved Morris's result and obtained similar result for a large class of superlinear function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq15_HTML.gif .

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq16_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ2_HTML.gif
(1.2)

that is, the differential equation (1.1) issemilinear, similar results also hold, but the proof is more difficult since there may be resonant case. We refer to [68] and the references therein.

In [8] Liu considered the following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq17_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq19_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq20_HTML.gif -periodic function. After introducing new variables, the differential equation (1.3) can be changed into a Hamiltonian system. Under some suitable assumptions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq22_HTML.gif , by using a variant of Moser's small twist theorem [9] to the Pioncaré map, the author obtained the existence of quasi-periodic solutions and the boundedness of all solutions.

Then the result is generalized to a class of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq23_HTML.gif -Laplacian differential equation.Yang [10] considered the following nonlinear differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq24_HTML.gif is bounded, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq25_HTML.gif is periodic. The idea is also to change the original problem to Hamiltonian system and then use a twist theorem of area-preserving mapping to the Pioncaré map.

The above differential equation essentially possess Hamiltonian structure. It is well known that the Hamiltonian structure and reversible structure have many similar property. Especially, they have similar KAM theorem.

Recently, Liu [6] studied the following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ5_HTML.gif
(1.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq26_HTML.gif is a positive constant and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq27_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq28_HTML.gif -periodic in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq29_HTML.gif . Under some assumption of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq31_HTML.gif , the differential equation (1.5) has a reversible structure. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq32_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ6_HTML.gif
(1.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq34_HTML.gif . Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ7_HTML.gif
(1.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq35_HTML.gif is a constant. Note that here and below we always use https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq36_HTML.gif to indicate some constants. Assume that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq37_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ8_HTML.gif
(1.8)

Then, the following conclusions hold true.

(i)There exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq38_HTML.gif and a closed set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq39_HTML.gif having positive measure such that for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq40_HTML.gif , there exists a quasi-periodic solution for (1.5) with the basic frequency https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq41_HTML.gif .

(ii)Every solution of (1.5) is bounded.

Motivated by the papers [6, 8, 10], we consider the following https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq42_HTML.gif -Laplacian equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ9_HTML.gif
(1.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq43_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq45_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq46_HTML.gif are constants. We want to generalize the result in [6] to a class of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq47_HTML.gif -Laplacian-type differential equations of the form (1.9). The main idea is similar to that in [6]. We will assume that the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq49_HTML.gif have some parities such that the differential system (1.9) still has a reversible structure. After some transformations, we change the systems (1.9) to a form of small perturbation of integrable reversible system. Then a KAM Theorem for reversible mapping can be applied to the Poincaré mapping of this nearly integrable reversible system and some desired result can be obtained.

Our main result is the following theorem.

Theorem 1.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq51_HTML.gif are of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq52_HTML.gif in their arguments and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq53_HTML.gif -periodic with respect to t such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ10_HTML.gif
(1.10)
Moreover, suppose that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq54_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ11_HTML.gif
(1.11)

for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq55_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq57_HTML.gif . Then every solution of (1.9) is bounded.

Remark 1.2.

Our main nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq58_HTML.gif in (1.9) corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq59_HTML.gif in (1.5). Although it is more special than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq60_HTML.gif , it makes no essential difference of proof and can simplify our proof greatly. It is easy to see from the proof that this main nonlinearity is used to guarantee the small twist condition.

2. The Proof of Theorem

The proof of Theorem 1.1 is based on Moser's small twist theorem for reversible mapping. It mainly consists of two steps. The first one is to find an equivalent system, which has a nearly integrable form of a reversible system. The second one is to show that Pincaré map of the equivalent system satisfies some twist theorem for reversible mapping.

2.1. Action-Angle Variables

We first recall the definitions of reversible system. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq61_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq62_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq63_HTML.gif be an open domain, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq64_HTML.gif be continuous. Suppose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq65_HTML.gif is an involution (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq66_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq67_HTML.gif -diffeomorphism such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq68_HTML.gif ) satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq69_HTML.gif . The differential equations system
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ12_HTML.gif
(2.1)
is called reversible with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq70_HTML.gif , if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ13_HTML.gif
(2.2)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq71_HTML.gif denoting the Jacobian matrix of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq72_HTML.gif .

We are interested in the special involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq73_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq74_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq75_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq76_HTML.gif is reversible with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq77_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ14_HTML.gif
(2.3)

Below we will see that the symmetric properties (1.10) imply a reversible structure of the system (1.9).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq78_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq79_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq80_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq81_HTML.gif . Hence, the differential equation (1.9) is changed into the following planar system:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ15_HTML.gif
(2.4)

By (1.10) it is easy to see that the system (2.4) is reversible with respect to the involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq82_HTML.gif .

Below we will write the reversible system (2.4) as a form of small perturbation. For this purpose we first introduce action-angle variables https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq83_HTML.gif .

Consider the homogeneous differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ16_HTML.gif
(2.5)
This equation takes as an integrable part of (1.9). We will use its solutions to construct a pair of action-angle variables. One of solutions for (2.5) is the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq84_HTML.gif as defined below. Let the number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq85_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ17_HTML.gif
(2.6)
We define the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq86_HTML.gif , implicitly by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ18_HTML.gif
(2.7)

The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq87_HTML.gif will be extended to the whole real axis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq88_HTML.gif as explained below, and the extension will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq89_HTML.gif . Define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq90_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq91_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq92_HTML.gif . Then, we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq93_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq94_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq95_HTML.gif is an odd function. Finally, we extend https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq96_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq97_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq98_HTML.gif -periodicity. It is not difficult to verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq99_HTML.gif has the following properties:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq100_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq101_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq102_HTML.gif ;

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq103_HTML.gif is an odd periodic function with period https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq104_HTML.gif .

It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq105_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ19_HTML.gif
(2.8)
with initial condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq106_HTML.gif . Define a transformation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq107_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ20_HTML.gif
(2.9)
It is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ21_HTML.gif
(2.10)
Since the Jacobian matrix of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq108_HTML.gif is nonsingular for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq109_HTML.gif , the transformation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq110_HTML.gif is a local homeomorphism at each point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq111_HTML.gif of the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq112_HTML.gif , while https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq113_HTML.gif is a global homeomorphism from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq114_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq115_HTML.gif . Under the transformation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq116_HTML.gif the system (2.4) is changed to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ22_HTML.gif
(2.11)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ23_HTML.gif
(2.12)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq117_HTML.gif .

It is easily verified that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq119_HTML.gif and so the system (2.11) is reversible with respect to the involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq120_HTML.gif .

2.2. Some Lemmas

To estimate https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq122_HTML.gif , we need some definitions and lemmas.

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq123_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq125_HTML.gif satisfy (1.11), then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ24_HTML.gif
(2.13)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq127_HTML.gif .

Proof.

We only prove the second inequality since the first one can be proved similarly.
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ25_HTML.gif
(2.14)

To describe the estimates in Lemma 2.1, we introduce function space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq128_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq129_HTML.gif is a function of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq130_HTML.gif .

Definition 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq131_HTML.gif . We say https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq132_HTML.gif , if for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq133_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq135_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ26_HTML.gif
(2.15)

Lemma 2.3 (see [6]).

The following conclusions hold true:

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq136_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq137_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq138_HTML.gif ;

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq140_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq141_HTML.gif ;

(iii)Suppose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq142_HTML.gif satisfy that, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq143_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq144_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ27_HTML.gif
(2.16)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq145_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq146_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq147_HTML.gif , then, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ28_HTML.gif
(2.17)
Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ29_HTML.gif
(2.18)

Proof.

This lemma was proved in [6], but we give the proof here for reader's convenience. Since (i) and (ii) are easily verified by definition, so we only prove (iii). Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ30_HTML.gif
(2.19)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq148_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq149_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq150_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq151_HTML.gif is bounded and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq152_HTML.gif . Similarly, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ31_HTML.gif
(2.20)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq153_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ32_HTML.gif
(2.21)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq154_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ33_HTML.gif
(2.22)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq155_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq156_HTML.gif , we know that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq157_HTML.gif sufficiently large
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ34_HTML.gif
(2.23)
By the property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq158_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ35_HTML.gif
(2.24)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq159_HTML.gif sufficiently large.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq160_HTML.gif , then by a direct computation, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ36_HTML.gif
(2.25)
where the sum is found for the indices satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ37_HTML.gif
(2.26)
Without loss of generality, we assume that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ38_HTML.gif
(2.27)

Furthermore, we suppose that among https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq161_HTML.gif , there are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq162_HTML.gif numbers which equal to 0, and among https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq163_HTML.gif , there are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq164_HTML.gif numbers which equal to 0.

Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ39_HTML.gif
(2.28)
we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ40_HTML.gif
(2.29)
and then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ41_HTML.gif
(2.30)
Obviously
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ42_HTML.gif
(2.31)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ43_HTML.gif
(2.32)
By the condition of (iii) we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ44_HTML.gif
(2.33)

In the same way we can consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq165_HTML.gif and we omit the details.

2.3. Some Estimates

The following lemma gives the estimate for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq167_HTML.gif .

Lemma 2.4.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq169_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq170_HTML.gif .

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq171_HTML.gif , we first consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq173_HTML.gif . By Lemma 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq174_HTML.gif . Again https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq175_HTML.gif , using the conclusion (iii) of Lemma 2.3, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq176_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq177_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq179_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq180_HTML.gif . In the same way we can prove https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq181_HTML.gif . Thus Lemma 2.4 is proved.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq182_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq183_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq184_HTML.gif for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq185_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq186_HTML.gif the system (2.11) is equivalent to the following system:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ45_HTML.gif
(2.34)

It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq187_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq188_HTML.gif . Hence, system (2.34) is reversible with respect to the involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq189_HTML.gif .

We will prove that the Poincaré mapping can be a small perturbation of integrable reversible mapping. For this purpose, we write (2.34) as a small perturbation of an integrable reversible system. Write the system (2.34) in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ46_HTML.gif
(2.35)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq190_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq191_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq193_HTML.gif defined in (2.11). It follows https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq194_HTML.gif , and so (2.35) is also reversible with respect to the involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq195_HTML.gif . Below we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq197_HTML.gif are smaller perturbations.

Lemma 2.5.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq199_HTML.gif .

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq200_HTML.gif is sufficiently large, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq201_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq202_HTML.gif . Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ47_HTML.gif
(2.36)
It is easy to verify that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ48_HTML.gif
(2.37)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq203_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq204_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq205_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq206_HTML.gif are defined in the same way as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq207_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq208_HTML.gif .

So, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ49_HTML.gif
(2.38)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ50_HTML.gif
(2.39)
So
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ51_HTML.gif
(2.40)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq209_HTML.gif . In the same way, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq210_HTML.gif .

Now we change system (2.35) to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ52_HTML.gif
(2.41)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq211_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq212_HTML.gif . By the proof of Lemma 2.4, we know https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq214_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq216_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ53_HTML.gif
(2.42)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq217_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq218_HTML.gif .

2.4. Coordination Transformation

Lemma 2.6.

There exists a transformation of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ54_HTML.gif
(2.43)
such that the system (2.41) is changed into the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ55_HTML.gif
(2.44)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq219_HTML.gif satisfy:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ56_HTML.gif
(2.45)

Moreover, the system (2.44) is reversible with respect to the involution G: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq220_HTML.gif .

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ57_HTML.gif
(2.46)
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ58_HTML.gif
(2.47)
It is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ59_HTML.gif
(2.48)
Hence the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq221_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq222_HTML.gif is diffeomorphism for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq223_HTML.gif . Thus, there is a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq224_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ60_HTML.gif
(2.49)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ61_HTML.gif
(2.50)
Under this transformation, the system (2.41) is changed to (2.44) with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ62_HTML.gif
(2.51)

Below we estimate https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq226_HTML.gif . We only consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq227_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq228_HTML.gif can be considered similarly or even simpler.

Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ63_HTML.gif
(2.52)
Note that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ64_HTML.gif
(2.53)
By the third conclusion of Lemma 2.3, we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ65_HTML.gif
(2.54)
In the same way as the above, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ66_HTML.gif
(2.55)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ67_HTML.gif
(2.56)
By (2.54) and (2.56), noting that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq229_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ68_HTML.gif
(2.57)

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq230_HTML.gif , the system (2.44) is reversible in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq231_HTML.gif with respect to the involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq232_HTML.gif . Thus Lemma 2.6 is proved.

Now we make average on the nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq233_HTML.gif in the second equation of (2.44).

Lemma 2.7.

There exists a transformation of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ69_HTML.gif
(2.58)
which changes (2.44) to the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ70_HTML.gif
(2.59)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq234_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq235_HTML.gif and the new perturbations https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq236_HTML.gif satisfy:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ71_HTML.gif
(2.60)

Moreover, the system (2.59) is reversible with respect to the involution G: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq237_HTML.gif .

Proof.

We choose
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ72_HTML.gif
(2.61)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ73_HTML.gif
(2.62)
Defined a transformation by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ74_HTML.gif
(2.63)
Then the system of (2.44) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ75_HTML.gif
(2.64)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ76_HTML.gif
(2.65)
It is easy to very that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ77_HTML.gif
(2.66)
which implies that the system (2.59) is reversible with respect to the involution G: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq238_HTML.gif . In the same way as the proof of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq239_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq240_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ78_HTML.gif
(2.67)

Thus Lemma 2.7 is proved.

Below we introduce a small parameter such that the system (2.4) is written as a form of small perturbation of an integrable.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ79_HTML.gif
(2.68)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ80_HTML.gif
(2.69)
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ81_HTML.gif
(2.70)
Now, we define a transformation by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ82_HTML.gif
(2.71)
Then the system (2.59) has the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ83_HTML.gif
(2.72)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ84_HTML.gif
(2.73)

Lemma 2.8.

The perturbations https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq241_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq242_HTML.gif satisfy the following estimates:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ85_HTML.gif
(2.74)

Proof.

By (2.73), (2.60) and noting that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq243_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ86_HTML.gif
(2.75)

In the same way, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq244_HTML.gif . The estimates (2.74) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq245_HTML.gif follow easily from (2.60).

2.5. Poincaré Map and Twist Theorems for Reversible Mapping

We can use a small twist theorem for reversible mapping to prove that the Pioncaré map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq246_HTML.gif has an invariant closed curve, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq247_HTML.gif is sufficiently small. The earlier result was due to Moser [11, 12], and Sevryuk [13]. Later, Liu [14] improved the previous results. Let us first recall the theorem in [14].

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq248_HTML.gif be a finite part of cylinder https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq249_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq250_HTML.gif , we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq251_HTML.gif the class of Jordan curves in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq252_HTML.gif that are homotopic to the circle https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq253_HTML.gif . The subclass of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq254_HTML.gif composed of those curves lying in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq255_HTML.gif will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq256_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ87_HTML.gif
(2.76)
Consider a mapping https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq257_HTML.gif , which is reversible with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq258_HTML.gif . Moreover, a lift of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq259_HTML.gif can be expressed in the form:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ88_HTML.gif
(2.77)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq260_HTML.gif is a real number, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq261_HTML.gif is a small parameter, the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq262_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq263_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq264_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq265_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq266_HTML.gif periodic.

Lemma 2.9 (see [14, Theorem 2]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq267_HTML.gif with an integer n and the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq268_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq269_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq270_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq271_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ89_HTML.gif
(2.78)
In addition, we assume that there is a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq272_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ90_HTML.gif
(2.79)
Moreover, suppose that there are two numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq273_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq274_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq275_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ91_HTML.gif
(2.80)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ92_HTML.gif
(2.81)
Then there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq277_HTML.gif such that, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq278_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ93_HTML.gif
(2.82)

the mapping https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq279_HTML.gif has an invariant curve in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq280_HTML.gif , the constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq281_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq282_HTML.gif depend on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq283_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq284_HTML.gif . In particular, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq285_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq286_HTML.gif .

Remark 2.10.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq287_HTML.gif satisfy all the conditions of Lemma 2.9, then Lemma 2.9 still holds.

Lemma 2.11 (see [14, Theorem 1]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq288_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq289_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq290_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq291_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq292_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ94_HTML.gif
(2.83)
then there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq293_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq294_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq295_HTML.gif has an invariant curve in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq296_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq297_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ95_HTML.gif
(2.84)

The constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq298_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq299_HTML.gif depend on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq300_HTML.gif only.

We use Lemmas 2.9 and 2.11 to prove our Theorem 1.1. For the reversible mapping (2.86), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq301_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq302_HTML.gif .

2.6. Invariant Curves

From (2.73) and (2.66), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ96_HTML.gif
(2.85)
which yields that system (2.72) is reversible in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq303_HTML.gif with respect to the involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq304_HTML.gif . Denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq305_HTML.gif the Poincare map of (2.72), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq306_HTML.gif is also reversible with the same involution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq307_HTML.gif and has the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ97_HTML.gif
(2.86)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq308_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq309_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq310_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq311_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ98_HTML.gif
(2.87)

Case 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq312_HTML.gif is rational).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq313_HTML.gif , it is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ99_HTML.gif
(2.88)

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq314_HTML.gif only depends on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq315_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq316_HTML.gif , all conditions in Lemma 2.9 hold.

Case 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq317_HTML.gif is irrational).

Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ100_HTML.gif
(2.89)

all the assumptions in Lemma 2.11 hold.

Thus, in the both cases, the Poincare mapping https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq318_HTML.gif always have invariant curves for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq319_HTML.gif being sufficient small. Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq320_HTML.gif , we know that for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq321_HTML.gif , there is an invariant curve of the Poincare mapping, which guarantees the boundedness of solutions of the system (2.11). Hence, all the solutions of (1.9) are bounded.

Authors’ Affiliations

(1)
Department of Mathematics, Southeast University

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© Shunjun Jiang et al. 2011

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