Open Access

On Step-Like Contrast Structure of Singularly Perturbed Systems

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]

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq1_HTML.gif is a small parameter and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq3_HTML.gif has two isolated solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq4_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq5_HTML.gif , which satisfy the following condition:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq6_HTML.gif in the phase plane https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq7_HTML.gif of the associated equations given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ3_HTML.gif
(13)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq8_HTML.gif is fixed and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq11_HTML.gif . The principal value https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq12_HTML.gif of an internal transition time https://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:

https://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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq14_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq16_HTML.gif of the reduced equations https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq17_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq18_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq20_HTML.gif of the associated equations given by

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

satisfy the condition as follows:

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq28_HTML.gif and https://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 https://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 https://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:
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ11_HTML.gif
(22)

where https://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 https://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 https://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,

https://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
https://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.

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

https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ14_HTML.gif
(25)

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ15_HTML.gif
(26)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ16_HTML.gif
(27)
has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq43_HTML.gif real valued solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq45_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ17_HTML.gif
(28)

Remark 2.1.

[ https://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
https://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 [ https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq47_HTML.gif ], there may exist a solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq49_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq52_HTML.gif such that the following limit holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ19_HTML.gif
(210)
We regard the solution https://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: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq57_HTML.gif . That is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ20_HTML.gif
(211)
The assumption [ https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ21_HTML.gif
(212)

has two equilibria https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq59_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq60_HTML.gif ), where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq62_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq63_HTML.gif dimensions and an unstable manifold https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq65_HTML.gif . To get a heteroclinic orbit connecting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq66_HTML.gif and https://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.

[ https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ22_HTML.gif
(213)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq69_HTML.gif is an arbitrary constant and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq71_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq72_HTML.gif ) can be represented by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ23_HTML.gif
(214)
[ https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq74_HTML.gif , which is denoted by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ24_HTML.gif
(215)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq75_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq76_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq78_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq80_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ25_HTML.gif
(216)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ26_HTML.gif
(217)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ27_HTML.gif
(218)

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq82_HTML.gif and it yields a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq83_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq84_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq86_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ28_HTML.gif
(219)

is solvable with respect to https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ29_HTML.gif
(31)

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

The position of a transition time https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ30_HTML.gif
(32)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq110_HTML.gif ( https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ31_HTML.gif
(33)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq112_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq113_HTML.gif ) are all constants, independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq114_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq115_HTML.gif takes value between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq117_HTML.gif . For example, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ32_HTML.gif
(34)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq120_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq121_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq126_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq128_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq130_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq132_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq133_HTML.gif ) are given by

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq134_HTML.gif , (3.5) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ34_HTML.gif
(36)
The equations to determine https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq135_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq136_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq137_HTML.gif ) are given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ35_HTML.gif
(37)

Here the superscript https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq138_HTML.gif is omitted for the variables https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq139_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq142_HTML.gif , we agree that they take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq143_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq144_HTML.gif ; while they take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq145_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq146_HTML.gif . The terms https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq147_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq148_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq149_HTML.gif ) are expressed in terms of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq151_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq152_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq153_HTML.gif ). Also https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq155_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq156_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq158_HTML.gif when https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq160_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq161_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq162_HTML.gif ) is uniquely solvable by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq164_HTML.gif as follows:
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ38_HTML.gif
(310)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ39_HTML.gif
(311)

From https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq165_HTML.gif , it yields that the equilibrium https://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 https://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.

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

Then, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq173_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq174_HTML.gif ) are known values after https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq175_HTML.gif being solved by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq177_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ40_HTML.gif
(312)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ42_HTML.gif
(314)
https://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.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq178_HTML.gif Suppose that the hypercurve https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq179_HTML.gif intersects the manifold https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq180_HTML.gif in the phase space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq181_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq184_HTML.gif , the solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq185_HTML.gif ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq187_HTML.gif is determined, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq188_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq189_HTML.gif ) are completely known. To determine https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq191_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ44_HTML.gif
(316)
If we show https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq195_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ45_HTML.gif
(317)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq196_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq197_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq198_HTML.gif ) only depend on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq199_HTML.gif , while https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq200_HTML.gif only depends on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq203_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq204_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ46_HTML.gif
(318)
or
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq205_HTML.gif . Then, by [ https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq206_HTML.gif ], there exists an https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq210_HTML.gif ( https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ48_HTML.gif
(320)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq213_HTML.gif ; https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq215_HTML.gif ; and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq216_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq218_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq220_HTML.gif ( https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ49_HTML.gif
(321)

Then, https://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 https://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 https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ50_HTML.gif
(41)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ51_HTML.gif
(42)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq225_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq226_HTML.gif is a parameter, such a solution https://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 [ https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq228_HTML.gif ]–[ https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq229_HTML.gif ] [14, 15]. Then, we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq231_HTML.gif .

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq232_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq235_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ54_HTML.gif
(45)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ55_HTML.gif
(46)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq237_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq239_HTML.gif indeed in the neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq240_HTML.gif such that the solution https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq244_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq246_HTML.gif , the boundary functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq247_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq251_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq252_HTML.gif
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ58_HTML.gif
(49)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq253_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ60_HTML.gif
(411)
After the transformation given by
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ62_HTML.gif
(413)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ63_HTML.gif
(414)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq254_HTML.gif and https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq256_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq257_HTML.gif ; and there exists a first integral
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq258_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq259_HTML.gif .

In views of https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq261_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq262_HTML.gif ; and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq263_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq264_HTML.gif for the solutions https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ67_HTML.gif
(418)
Let
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_Equ69_HTML.gif
(420)

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

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

Then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq273_HTML.gif such that https://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 [ https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq275_HTML.gif ]–[ https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq276_HTML.gif ] hold. Then, there exists an https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq278_HTML.gif ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq280_HTML.gif . Moreover, the following asymptotic expansion holds
https://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 [ https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq281_HTML.gif ]–[ https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F634324/MediaObjects/13661_2009_Article_867_IEq283_HTML.gif and https://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

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