On Step-Like Contrast Structure of Singularly Perturbed Systems

  • Mingkang Ni1 and

    Affiliated with

    • Zhiming Wang1Email author

      Affiliated with

      Boundary Value Problems20092009:634324

      DOI: 10.1155/2009/634324

      Received: 15 April 2009

      Accepted: 14 July 2009

      Published: 23 August 2009

      Abstract

      The existence of a step-like contrast structure for a class of high-dimensional singularly perturbed system is shown by a smooth connection method based on the existence of a first integral for an associated system. In the framework of this paper, we not only give the conditions under which there exists an internal transition layer but also determine where an internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of a solution with a step-like contrast structure is presented.

      1. Introduction

      The problem of contrast structures is a singularly perturbed problem whose solutions with both internal transition layers and boundary layers. In recent years, the study of contrast structures is one of the hot research topics in the study of singular perturbation theory. In western society, most works on internal layer solutions concentrate on singularly perturbed parabolic systems by geometric method (see [1] and the references therein). In Russia, the works on singularly perturbed ordinary equations are concerned by boundary function method [25]. One of the basic difficulties for such a problem is unknown of where an internal transition layer is in advance.

      Butuzov and Vasil'eva initiated the concept of contrast structures in 1987 [6] and studied the following boundary value problem of a second-order semilinear equation with a step-like contrast structure, which is called a monolayer solution in [1]

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ1_HTML.gif
      (11)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq1_HTML.gif is a small parameter and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq2_HTML.gif has a desired smooth scalar function on its arguments.

      Suppose that the reduced equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq3_HTML.gif has two isolated solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq4_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq5_HTML.gif , which satisfy the following condition:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ2_HTML.gif
      (12)
      The condition (1.2) indicates that there exist two saddle equilibria http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq6_HTML.gif in the phase plane http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq7_HTML.gif of the associated equations given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ3_HTML.gif
      (13)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq8_HTML.gif is fixed and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq9_HTML.gif .

      It is shown in [6] that the existence of an internal transition layer for the problem (1.1) is closely related to the existence of a heteroclinic orbit connecting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq11_HTML.gif . The principal value http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq12_HTML.gif of an internal transition time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq13_HTML.gif is determined by an equation as follows:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ4_HTML.gif
      (14)

      In [7], Vasil'eva further studied the existence of step-like contrast structures for a class of singularly perturbed equations given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ5_HTML.gif
      (15)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq15_HTML.gif are scalar functions. For (1.5), we may impose either a first class of boundary condition or a second class of boundary condition.

      Suppose that there exist two solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq16_HTML.gif of the reduced equations http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq17_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq18_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq19_HTML.gif are two saddle equilibria in the phase plane http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq20_HTML.gif of the associated equations given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ6_HTML.gif
      (16)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq21_HTML.gif is fixed with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq22_HTML.gif . This indicates that the eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq23_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq24_HTML.gif ) of the Jacobian matrix
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ7_HTML.gif
      (17)

      satisfy the condition as follows:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ8_HTML.gif
      (18)

      If (1.6) is a Hamilton equation, that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq25_HTML.gif , it implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq26_HTML.gif . Then, the equation to determine http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq27_HTML.gif is given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ9_HTML.gif
      (19)

      Geometrically, (1.9) is also a condition for the existence of a heteroclinic orbit connecting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq29_HTML.gif .

      Unfortunately, for a high dimensional singularly perturbed system, we cannot always find such an equation like (1.9) to determine http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq30_HTML.gif at which there exists a heteroclinic orbit. This is one difficulty to further study the problem on step-like contrast structures. On the other hand, we know that the existence of a spike-like or a step-like contrast structure of high dimension is closely related to the existence of a homoclinic or heteroclinic orbit in its corresponding phase space, respectively. However, the existence of a homoclinic or heteroclinic orbit in high dimension space and how to construct such an orbit are themselves open in general in the qualitative analysis (geometric method) theory [810]. To explore these high dimensional contrast structure problems, we just start from some particular class of singularly perturbed system and are trying to develop some approach to construct a desired heteroclinic orbit by using a first integral method for such a class of the system and determine its internal transition time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq31_HTML.gif .

      2. Problem Formulation

      We consider a class of semilinear singularly perturbed system as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ10_HTML.gif
      (21)
      with a first class of boundary condition given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ11_HTML.gif
      (22)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq32_HTML.gif is a small parameter.

      The class of system (2.1) in question has a strong application background in engineering. For example, in the study of smart materials of variated current of liquid [11, 12], its math model is a kind of such a system like (2.1), where the small parameter http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq33_HTML.gif indicates a particle. The given boundary condition (2.2) corresponds the stability condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq34_HTML.gif listed later to ensure that there exists a solution for the problem in question.

      For our convenience, the system (2.1) can also be written in the following equivalent form,

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ12_HTML.gif
      (23)
      Then, the corresponding boundary condition (2.2) is now written as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ13_HTML.gif
      (24)

      The following assumptions are fundamental in theory for the problem in question.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq35_HTML.gif Suppose that the functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq36_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq37_HTML.gif ) are sufficiently smooth on the domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq38_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq39_HTML.gif are real numbers.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq40_HTML.gif Suppose that the reduced system of (2.1) given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ14_HTML.gif
      (25)

      has two isolated solutions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq41_HTML.gif :

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ15_HTML.gif
      (26)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq42_HTML.gif Suppose that the characteristic equation of the system (2.3) given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ16_HTML.gif
      (27)
      has http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq43_HTML.gif real valued solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq45_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ17_HTML.gif
      (28)

      Remark 2.1.

      [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq46_HTML.gif ] is called as a stability condition. For a more general stability condition given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ18_HTML.gif
      (29)

      it will be studied in the other paper because of more complicated dynamic performance presented.

      Under the assumption of [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq47_HTML.gif ], there may exist a solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq48_HTML.gif with only two boundary layers that occurred at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq50_HTML.gif , for which the detailed discussion has been given by [13, Theorem  4.2], or it may consults [5, Theorem  2.4, page 49]. We are only interested in a solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq51_HTML.gif with a step-like contrast structure in this paper. That is, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq52_HTML.gif such that the following limit holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ19_HTML.gif
      (210)
      We regard the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq53_HTML.gif defined above with such a step-like contrast structure as being smoothly connected by two pure boundary solutions: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq56_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq57_HTML.gif . That is,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ20_HTML.gif
      (211)
      The assumption [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq58_HTML.gif ] ensures that the corresponding associated system given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ21_HTML.gif
      (212)

      has two equilibria http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq59_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq60_HTML.gif ), where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq61_HTML.gif is fixed. They are both hyperbolic saddle points. From [13, Theorem  4.2] (or [5, Theorem  2.4]), it yields that there exists a stable manifold http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq62_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq63_HTML.gif dimensions and an unstable manifold http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq64_HTML.gif of one-dimension in a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq65_HTML.gif . To get a heteroclinic orbit connecting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq67_HTML.gif in the corresponding phase space, we need some more assumptions as follows.

      [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq68_HTML.gif ] Suppose that the associated system (2.12) has a first integral
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ22_HTML.gif
      (213)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq69_HTML.gif is an arbitrary constant and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq70_HTML.gif is a smooth function on its arguments.

      Then, the first integral passing through http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq71_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq72_HTML.gif ) can be represented by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ23_HTML.gif
      (214)
      [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq73_HTML.gif ] Suppose that (2.14) is solvable with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq74_HTML.gif , which is denoted by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ24_HTML.gif
      (215)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq77_HTML.gif be the parametric expressions of orbit passing through the hyperbolic saddle points http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq79_HTML.gif , respectively.

      Corresponding to the given boundary condition (2.2), we consider the following initial value relation at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq80_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ25_HTML.gif
      (216)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ26_HTML.gif
      (217)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ27_HTML.gif
      (218)

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq81_HTML.gif Suppose that (2.17) is solvable with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq82_HTML.gif and it yields a solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq83_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq85_HTML.gif .

      Remark 2.2.

      It is easy to see from (2.14) and (2.17) that the necessary condition of the existence of a heteroclinic orbit connecting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq87_HTML.gif can also be expressed as "the equation
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ28_HTML.gif
      (219)

      is solvable with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq88_HTML.gif .''

      3. Construction of Asymptotic Solution

      We seek an asymptotic solution of the problem (2.1)-(2.2) of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ29_HTML.gif
      (31)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq89_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq91_HTML.gif ; and for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq92_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq93_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq94_HTML.gif ) are coefficients of regular terms; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq95_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq96_HTML.gif ) are coefficients of boundary layer terms at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq97_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq98_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq99_HTML.gif ) are coefficients of boundary layer terms at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq100_HTML.gif ; and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq101_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq102_HTML.gif ) are left and right coefficients of internal transition terms at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq103_HTML.gif . Meanwhile, similar definitions are for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq104_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq105_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq107_HTML.gif .

      The position of a transition time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq108_HTML.gif is unknown in advance. It needs being determined during the construction of an asymptotic solution. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq109_HTML.gif has also an asymptotic expression of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ30_HTML.gif
      (32)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq110_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq111_HTML.gif ) are temporarily unknown at the moment and will be determined later.

      Meanwhile, let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ31_HTML.gif
      (33)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq112_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq113_HTML.gif ) are all constants, independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq114_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq115_HTML.gif takes value between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq116_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq117_HTML.gif . For example, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq118_HTML.gif .

      Then, we will determine the asymptotic solution (3.1) step by step using "a smooth connection method'' based on the boundary function method [13] or [5]. The smooth connection condition (2.11) can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ32_HTML.gif
      (34)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq120_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq122_HTML.gif are all the known functions depending only on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq123_HTML.gif .

      Substituting (3.1) into (2.1)-(2.2) and equating separately the terms depending on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq124_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq127_HTML.gif by the boundary function method, we can obtain the equations to determine http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq128_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq131_HTML.gif , respectively. The equations to determine the zero-order coefficients of regular terms http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq132_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq133_HTML.gif ) are given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ33_HTML.gif
      (35)
      It is clear to see that (3.5) coincides with the reduced system (2.11). Therefore, by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq134_HTML.gif , (3.5) has the solution
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ34_HTML.gif
      (36)
      The equations to determine http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq136_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq137_HTML.gif ) are given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ35_HTML.gif
      (37)

      Here the superscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq138_HTML.gif is omitted for the variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq140_HTML.gif in (3.7) for simplicity in notation. To understand http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq142_HTML.gif , we agree that they take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq143_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq144_HTML.gif ; while they take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq145_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq146_HTML.gif . The terms http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq147_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq148_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq149_HTML.gif ) are expressed in terms of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq151_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq152_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq153_HTML.gif ). Also http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq154_HTML.gif are known functions that take value at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq155_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq156_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq158_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq159_HTML.gif .

      Since (3.7) is an algebraic linear system, the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq160_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq161_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq162_HTML.gif ) is uniquely solvable by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq163_HTML.gif .

      Next, we give the equations and their conditions for determining the zero-order coefficient of an internal transition layer http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq164_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ36_HTML.gif
      (38)
      We rewrite (3.8) in a different form by making the change of variables
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ37_HTML.gif
      (39)
      Then, (3.8) is further written in these new variables as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ38_HTML.gif
      (310)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ39_HTML.gif
      (311)

      From http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq165_HTML.gif , it yields that the equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq166_HTML.gif of the autonomous system (3.10) is a hyperbolic saddle point. Therefore, there exists an unstable one-dimensional manifold http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq167_HTML.gif . For the existence of a solution of (3.10) satisfying (3.11), we need the following assumption.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq168_HTML.gif Suppose that the hyperplane http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq169_HTML.gif intersects the manifold http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq170_HTML.gif in the phase space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq171_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq172_HTML.gif is a parameter.

      Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq173_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq174_HTML.gif ) are known values after http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq175_HTML.gif being solved by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq176_HTML.gif . We can get the equations and the corresponding boundary conditions to determine http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq177_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ40_HTML.gif
      (312)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ41_HTML.gif
      (313)
      Introducing a similar transformation as doing for (3.8), we can get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ42_HTML.gif
      (314)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ43_HTML.gif
      (315)

      To ensure that the existence of a solution of (3.14)-(3.15), we need the following assumption.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq178_HTML.gif Suppose that the hypercurve http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq179_HTML.gif intersects the manifold http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq180_HTML.gif in the phase space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq181_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq182_HTML.gif is a parameter.

      Here it should be emphasized that under the conditions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq183_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq184_HTML.gif , the solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq185_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq186_HTML.gif ) not only exist but also decay exponentially [13], or [5].

      If the parameter http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq187_HTML.gif is determined, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq188_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq189_HTML.gif ) are completely known. To determine http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq190_HTML.gif , it is closely related to the existence of a heteroclinic orbit connecting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq192_HTML.gif in the phase space.

      By the given initial values (3.13) or (3.15), we have already obtained
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ44_HTML.gif
      (316)
      If we show http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq193_HTML.gif , the smooth connection condition (2.11) for the zero-order is satisfied. By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq194_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq195_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ45_HTML.gif
      (317)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq196_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq197_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq198_HTML.gif ) only depend on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq199_HTML.gif , while http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq200_HTML.gif only depends on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq201_HTML.gif , the necessary condition for existence of a heteroclinic orbit connecting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq203_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq204_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ46_HTML.gif
      (318)
      or
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ47_HTML.gif
      (319)

      However (3.18) or (3.19) is the one to determine http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq205_HTML.gif . Then, by [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq206_HTML.gif ], there exists an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq207_HTML.gif from (3.18) or (3.19). We can see that the process of determining http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq208_HTML.gif is the one of a smooth connection. Therefore, all the zero-order terms http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq209_HTML.gif have now been completely determined by the smooth connection for the zero-order coefficients of the asymptotic solution.

      For the high-order terms http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq210_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq211_HTML.gif ), we have the equations and their boundary conditions as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ48_HTML.gif
      (320)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq212_HTML.gif represent known functions that take value at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq213_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq214_HTML.gif are the known functions that only depend on those asymptotic terms whose subscript is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq215_HTML.gif ; and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq216_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq217_HTML.gif are all the known functions. Since (3.20) are all linear boundary value problems, it is not difficult to prove the existence of solution and the exponential decaying of solution without imposing any extra condition.

      As for boundary functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq218_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq219_HTML.gif , it is easy to obtain their constructions by using the normal boundary function method. So we would not discuss the details on them here [13] or [5]. However, it is worth mentioning that the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq220_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq221_HTML.gif ) in (2.3) will be determined by an equation as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ49_HTML.gif
      (321)

      Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq222_HTML.gif can be solved from (3.21) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq223_HTML.gif . Then, we have so far constructed the asymptotic expansion of a solution with an internal transition layer for the problem (2.1)-(2.2) and the asymptotic expansion of an internal transition time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq224_HTML.gif .

      4. Existence of Step-Like Solution and Its Limit Theorem

      We mentioned in Section 2that the solution with a step-like contrast structure can be regarded as a smooth connection by two solutions of pure boundary value problem from left and right, respectively. To this end, we establish the following two associated problems.

      For the left associated problem,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ50_HTML.gif
      (41)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ51_HTML.gif
      (42)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq225_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq226_HTML.gif is a parameter, such a solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq227_HTML.gif of (4.1) and (4.2) exists by [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq228_HTML.gif ]–[ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq229_HTML.gif ] [14, 15]. Then, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq230_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq231_HTML.gif .

      For the right associated problem,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ52_HTML.gif
      (43)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ53_HTML.gif
      (44)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq232_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq233_HTML.gif is still a parameter, the similar reason is for the existence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq234_HTML.gif of (4.3) and (4.4) [14, 15].

      Then, we write the asymptotic expansion of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq235_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ54_HTML.gif
      (45)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ55_HTML.gif
      (46)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq236_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq237_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq238_HTML.gif .

      We proceed to show that there exists an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq239_HTML.gif indeed in the neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq240_HTML.gif such that the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq241_HTML.gif of the left associated problem (4.1) and (4.2) and the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq242_HTML.gif of the right associated problem (4.3) and (4.4) smoothly connect at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq243_HTML.gif from which we obtain the desired step-like solution.

      From the asymptotic expansion of (4.5) and (4.6), we know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq244_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq245_HTML.gif are the solutions of the reduced system (2.5). In the neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq246_HTML.gif , the boundary functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq247_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq248_HTML.gif are both exponentially small. Thus, they can be omitted in the neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq249_HTML.gif .

      We are now concerned with the equations and the boundary conditions for which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq250_HTML.gif satisfy. They can be obtained easily from (3.8) just with the replacement of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq251_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq252_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ56_HTML.gif
      (47)
      After the change of variables given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ57_HTML.gif
      (48)
      then (4.7) can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ58_HTML.gif
      (49)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ59_HTML.gif
      (410)
      It is similar to get the equations and the boundary conditions for which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq253_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ60_HTML.gif
      (411)
      After the transformation given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ61_HTML.gif
      (412)
      then (4.11) can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ62_HTML.gif
      (413)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ63_HTML.gif
      (414)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq254_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq255_HTML.gif imply that there exists a first integral
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ64_HTML.gif
      (415)
      of the system (4.9) that approaches http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq256_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq257_HTML.gif ; and there exists a first integral
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ65_HTML.gif
      (416)

      of the system (4.13) that approaches http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq258_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq259_HTML.gif .

      In views of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq260_HTML.gif , from (4.15) and (4.16) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ66_HTML.gif
      (417)
      Then, we know from (4.2) and (4.4) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq261_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq262_HTML.gif ; and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq263_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq264_HTML.gif for the solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq265_HTML.gif of the left and right associated problems. For a smooth connection of the solutions, the remaining is to prove
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ67_HTML.gif
      (418)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ68_HTML.gif
      (419)
      Substituting (4.6) into (4.19), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ69_HTML.gif
      (420)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq266_HTML.gif can be regarded as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq267_HTML.gif for simplicity.

      If we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq268_HTML.gif in (4.20), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ70_HTML.gif
      (421)
      Since the sign of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq269_HTML.gif is fixed, (4.21) has an opposite sign when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq270_HTML.gif is sufficiently large, for example, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq271_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq272_HTML.gif is sufficiently small. That is,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ71_HTML.gif
      (422)

      Then, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq273_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq274_HTML.gif by applying the intermediate value theorem to (4.21). This implies in turn that (4.18) holds.

      Therefore, we have shown that there exists a step-like contrast structure for the problem (2.1)-(2.2). We summarize it as the following main theorem of this paper.

      Theorem 4.1.

      Suppose that [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq275_HTML.gif ]–[ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq276_HTML.gif ] hold. Then, there exists an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq277_HTML.gif such that there exists a step-like contrast structure solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq278_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq279_HTML.gif ) of the problem (2.1)-(2.2) when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq280_HTML.gif . Moreover, the following asymptotic expansion holds
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ72_HTML.gif
      (423)

      Remark 4.2.

      Only existence of solution with a step-like contrast structures is guarantied under the conditions of [ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq281_HTML.gif ]–[ http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq282_HTML.gif ]. There may exists a spike-like contrast structure, or the combination of them [16] for the problem (2.1)-(2.2). They need further study.

      5. Conclusive Remarks

      The existence of solution with step-like contrast structures for a class of high-dimensional singular perturbation problem investigated in this paper shows that how to get a heteroclinic orbit connecting saddle equilibria http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq283_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq284_HTML.gif in the corresponding phase space is a key to find a step-like internal layer solution. Using only one first integral of the associated system, this demands only bit information on solution, is our first try to construct a desired heteroclinic orbit in high-dimensional phase space. It needs surely further study for this interesting connection between the existence of a heteroclinic orbit of high-dimension in qualitative theory and the existence of a step-like contrast structure (internal layer solution) in a high-dimensional singular perturbation boundary value problem of ordinary differential equations. The particular boundary condition we adopt in this paper is just for the corresponding stability condition, which ensures the existence of solution of the problem in this paper. For the other type of boundary condition, we need some different stability condition to ensure the existence of solution of the problem in question, which we also need to study separately. Finally, if we want to construct a higher-order asymptotic expansion, it is similar with obvious modifications in which only more complicated techniques involved.

      Declarations

      Acknowledgments

      The authors are grateful for the referee's suggestion that helped to improve the presentation of this paper. This work was supported in part by E-Institutes of Shanghai Municipal Education Commission (N.E03004); and in part by the NSFC with Grant no. 10671069. This work was also supported by Shanghai Leading Academic Discipline Project with Project no. B407.

      Authors’ Affiliations

      (1)
      Department of Mathematics, East China Normal University

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      Copyright

      © M. Ni and Z. Wang 2009

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.