Lagrangian Stability of a Class of Second-Order Periodic Systems

  • Shunjun Jiang1,

    Affiliated with

    • Junxiang Xu1Email author and

      Affiliated with

      • Fubao Zhang1

        Affiliated with

        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: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq1_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq2_HTML.gif   ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq3_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq5_HTML.gif are positive constants, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq6_HTML.gif satisfies http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq8_HTML.gif and http://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
        http://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., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq10_HTML.gif as http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq12_HTML.gif , where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq15_HTML.gif .

        When http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq16_HTML.gif satisfies
        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ3_HTML.gif
        (1.3)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq17_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq19_HTML.gif is a http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq21_HTML.gif and http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ4_HTML.gif
        (1.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq24_HTML.gif is bounded, http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ5_HTML.gif
        (1.5)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq26_HTML.gif is a positive constant and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq27_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq28_HTML.gif -periodic in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq29_HTML.gif . Under some assumption of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq30_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq32_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ6_HTML.gif
        (1.6)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq34_HTML.gif . Moreover,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ7_HTML.gif
        (1.7)
        where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq37_HTML.gif such that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq38_HTML.gif and a closed set http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq42_HTML.gif -Laplacian equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ9_HTML.gif
        (1.9)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq43_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq45_HTML.gif , and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq48_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq50_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq51_HTML.gif are of class http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq52_HTML.gif in their arguments and http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq54_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ11_HTML.gif
        (1.11)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq55_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq56_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq58_HTML.gif in (1.9) corresponds to http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq61_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq62_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq63_HTML.gif be an open domain, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq64_HTML.gif be continuous. Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq65_HTML.gif is an involution (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq66_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq67_HTML.gif -diffeomorphism such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq68_HTML.gif ) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq69_HTML.gif . The differential equations system
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq70_HTML.gif , if
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ13_HTML.gif
        (2.2)

        with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq71_HTML.gif denoting the Jacobian matrix of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq73_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq74_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq75_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq76_HTML.gif is reversible with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq77_HTML.gif if and only if
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq78_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq79_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq80_HTML.gif satisfies http://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:
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq83_HTML.gif .

        Consider the homogeneous differential equation:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq84_HTML.gif as defined below. Let the number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq85_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ17_HTML.gif
        (2.6)
        We define the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq86_HTML.gif , implicitly by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ18_HTML.gif
        (2.7)

        The function http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq89_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq90_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq91_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq92_HTML.gif . Then, we define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq93_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq94_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq96_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq97_HTML.gif by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq99_HTML.gif has the following properties:

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

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

        (iii) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq104_HTML.gif .

        It is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq105_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ19_HTML.gif
        (2.8)
        with initial condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq106_HTML.gif . Define a transformation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq107_HTML.gif by
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq108_HTML.gif is nonsingular for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq109_HTML.gif , the transformation http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq111_HTML.gif of the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq112_HTML.gif , while http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq113_HTML.gif is a global homeomorphism from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq114_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq115_HTML.gif . Under the transformation http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ22_HTML.gif
        (2.11)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ23_HTML.gif
        (2.12)

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

        It is easily verified that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq118_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq120_HTML.gif .

        2.2. Some Lemmas

        To estimate http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq121_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq123_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq124_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq125_HTML.gif satisfy (1.11), then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ24_HTML.gif
        (2.13)

        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq126_HTML.gif , http://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.
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq128_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq129_HTML.gif is a function of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq130_HTML.gif .

        Definition 2.2.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq131_HTML.gif . We say http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq132_HTML.gif , if for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq133_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq135_HTML.gif such that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq136_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq137_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq138_HTML.gif ;

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

        (iii)Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq142_HTML.gif satisfy that, there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq143_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq144_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ27_HTML.gif
        (2.16)
        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq147_HTML.gif , then, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ28_HTML.gif
        (2.17)
        Moreover,
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ30_HTML.gif
        (2.19)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq148_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq149_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq150_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq151_HTML.gif is bounded and so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq152_HTML.gif . Similarly, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ31_HTML.gif
        (2.20)
        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq153_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ32_HTML.gif
        (2.21)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq154_HTML.gif , it follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ33_HTML.gif
        (2.22)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq155_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq156_HTML.gif , we know that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq157_HTML.gif sufficiently large
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ34_HTML.gif
        (2.23)
        By the property of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq158_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ35_HTML.gif
        (2.24)

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

        If http://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
        http://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
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq161_HTML.gif , there are http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq163_HTML.gif , there are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq164_HTML.gif numbers which equal to 0.

        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ39_HTML.gif
        (2.28)
        we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ40_HTML.gif
        (2.29)
        and then,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ41_HTML.gif
        (2.30)
        Obviously
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ42_HTML.gif
        (2.31)
        Since
        http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq166_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq167_HTML.gif .

        Lemma 2.4.

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

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq171_HTML.gif , we first consider http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq172_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq173_HTML.gif . By Lemma 2.1, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq174_HTML.gif . Again http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq176_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq177_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq179_HTML.gif , we have http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq182_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq183_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq184_HTML.gif for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq185_HTML.gif . When http://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:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq187_HTML.gif and http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ46_HTML.gif
        (2.35)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq190_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq191_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq192_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq193_HTML.gif defined in (2.11). It follows http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq195_HTML.gif . Below we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq196_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq197_HTML.gif are smaller perturbations.

        Lemma 2.5.

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

        Proof.

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq203_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq204_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq205_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq208_HTML.gif .

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

        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq209_HTML.gif . In the same way, we have http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ52_HTML.gif
        (2.41)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq211_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq214_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq216_HTML.gif where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ53_HTML.gif
        (2.42)

        with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq217_HTML.gif , http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ55_HTML.gif
        (2.44)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq219_HTML.gif satisfy:
        http://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: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq220_HTML.gif .

        Proof.

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ57_HTML.gif
        (2.46)
        then
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ59_HTML.gif
        (2.48)
        Hence the map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq221_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq222_HTML.gif is diffeomorphism for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq223_HTML.gif . Thus, there is a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq224_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ60_HTML.gif
        (2.49)
        where
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ62_HTML.gif
        (2.51)

        Below we estimate http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq225_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq226_HTML.gif . We only consider http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq227_HTML.gif since http://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,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ63_HTML.gif
        (2.52)
        Note that
        http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ66_HTML.gif
        (2.55)
        and so
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq229_HTML.gif , it follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ68_HTML.gif
        (2.57)

        Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq231_HTML.gif with respect to the involution http://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 http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ70_HTML.gif
        (2.59)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq234_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq235_HTML.gif and the new perturbations http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq236_HTML.gif satisfy:
        http://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: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq237_HTML.gif .

        Proof.

        We choose
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ72_HTML.gif
        (2.61)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ73_HTML.gif
        (2.62)
        Defined a transformation by
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ75_HTML.gif
        (2.64)
        where
        http://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
        http://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: http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq240_HTML.gif , we have
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ79_HTML.gif
        (2.68)
        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ80_HTML.gif
        (2.69)
        then
        http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ83_HTML.gif
        (2.72)
        where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq241_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq242_HTML.gif satisfy the following estimates:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq243_HTML.gif , it follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ86_HTML.gif
        (2.75)

        In the same way, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq244_HTML.gif . The estimates (2.74) for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq246_HTML.gif has an invariant closed curve, if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq248_HTML.gif be a finite part of cylinder http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq249_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq250_HTML.gif , we denote by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq251_HTML.gif the class of Jordan curves in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq252_HTML.gif that are homotopic to the circle http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq253_HTML.gif . The subclass of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq254_HTML.gif composed of those curves lying in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq255_HTML.gif will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq256_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ87_HTML.gif
        (2.76)
        Consider a mapping http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq257_HTML.gif , which is reversible with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq258_HTML.gif . Moreover, a lift of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq259_HTML.gif can be expressed in the form:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ88_HTML.gif
        (2.77)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq260_HTML.gif is a real number, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq261_HTML.gif is a small parameter, the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq262_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq263_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq264_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq265_HTML.gif are http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq268_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq269_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq270_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq271_HTML.gif satisfy
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq272_HTML.gif satisfying
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq273_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq274_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq275_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ91_HTML.gif
        (2.80)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ92_HTML.gif
        (2.81)
        Then there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq276_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq277_HTML.gif such that, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq278_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ93_HTML.gif
        (2.82)

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

        Remark 2.10.

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

        The constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq298_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq299_HTML.gif depend on http://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), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq301_HTML.gif , http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq303_HTML.gif with respect to the involution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq304_HTML.gif . Denote by http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq307_HTML.gif and has the form
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ97_HTML.gif
        (2.86)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq308_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq309_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq310_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq311_HTML.gif satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ98_HTML.gif
        (2.87)

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

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

        Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq314_HTML.gif only depends on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq315_HTML.gif , and http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq317_HTML.gif is irrational).

        Since
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq318_HTML.gif always have invariant curves for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq319_HTML.gif being sufficient small. Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_IEq320_HTML.gif , we know that for any http://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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