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On Step-Like Contrast Structure of Singularly Perturbed Systems
Boundary Value Problems volume 2009, Article number: 634324 (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 [2–5]. 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]
where 
is a small parameter and 
has a desired smooth scalar function on its arguments.
Suppose that the reduced equation 
has two isolated solutions 
on 
, which satisfy the following condition:
The condition (1.2) indicates that there exist two saddle equilibria 
in the phase plane 
of the associated equations given by
where 
is fixed and 
.
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 
and 
. The principal value 
of an internal transition time 
is determined by an equation as follows:
In [7], Vasil'eva further studied the existence of step-like contrast structures for a class of singularly perturbed equations given by
where 
and 
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 
of the reduced equations 
; 
, and 
are two saddle equilibria in the phase plane 
of the associated equations given by
where 
is fixed with 
. This indicates that the eigenvalues 
(
) of the Jacobian matrix
satisfy the condition as follows:
If (1.6) is a Hamilton equation, that is, 
, it implies that 
. Then, the equation to determine 
is given by
Geometrically, (1.9) is also a condition for the existence of a heteroclinic orbit connecting 
and 
.
Unfortunately, for a high dimensional singularly perturbed system, we cannot always find such an equation like (1.9) to determine 
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 [8–10]. 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 
.
2. Problem Formulation
We consider a class of semilinear singularly perturbed system as follows:
with a first class of boundary condition given by
where 
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 
indicates a particle. The given boundary condition (2.2) corresponds the stability condition 
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,
Then, the corresponding boundary condition (2.2) is now written as
The following assumptions are fundamental in theory for the problem in question.
Suppose that the functions 
(
) are sufficiently smooth on the domain 
, where 
are real numbers.
Suppose that the reduced system of (2.1) given by
has two isolated solutions on 
:
Suppose that the characteristic equation of the system (2.3) given by
has 
real valued solutions 
,
, where
Remark 2.1.
[
] is called as a stability condition. For a more general stability condition given by
it will be studied in the other paper because of more complicated dynamic performance presented.
Under the assumption of [
], there may exist a solution 
with only two boundary layers that occurred at 
and 
, 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 
with a step-like contrast structure in this paper. That is, there exists 
such that the following limit holds:
We regard the solution 
defined above with such a step-like contrast structure as being smoothly connected by two pure boundary solutions: 
,
and 
, 
. That is,
The assumption [
] ensures that the corresponding associated system given by
has two equilibria 
(
), where 
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 
of 
dimensions and an unstable manifold 
of one-dimension in a neighborhood of 
. To get a heteroclinic orbit connecting 
and 
in the corresponding phase space, we need some more assumptions as follows.
[
] Suppose that the associated system (2.12) has a first integral
where 
is an arbitrary constant and 
is a smooth function on its arguments.
Then, the first integral passing through 
(
) can be represented by
[
] Suppose that (2.14) is solvable with respect to 
, which is denoted by
Let 
and 
,
be the parametric expressions of orbit passing through the hyperbolic saddle points 
and 
, respectively.
Corresponding to the given boundary condition (2.2), we consider the following initial value relation at 
Let
where
Suppose that (2.17) is solvable with respect to 
and it yields a solution 
. That is, 
and 
.
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 
and 
can also be expressed as "the equation
is solvable with respect to 
.''
3. Construction of Asymptotic Solution
We seek an asymptotic solution of the problem (2.1)-(2.2) of the form
where 
, 
, 
; and for 
,
(
) are coefficients of regular terms; 
(
) are coefficients of boundary layer terms at 
; 
(
) are coefficients of boundary layer terms at 
; and 
(
) are left and right coefficients of internal transition terms at 
. Meanwhile, similar definitions are for 
, 
, 
and 
.
The position of a transition time 
is unknown in advance. It needs being determined during the construction of an asymptotic solution. Suppose that 
has also an asymptotic expression of the form
where 
(
) are temporarily unknown at the moment and will be determined later.
Meanwhile, let
where 
(
) are all constants, independent of 
, and 
takes value between 
and 
. For example, 
.
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
where 
,
; 
and 
are all the known functions depending only on 
.
Substituting (3.1) into (2.1)-(2.2) and equating separately the terms depending on 
, 
, 
and 
by the boundary function method, we can obtain the equations to determine 
; 
, 
and 
, respectively. The equations to determine the zero-order coefficients of regular terms 
(
) are given by
It is clear to see that (3.5) coincides with the reduced system (2.11). Therefore, by 
, (3.5) has the solution
The equations to determine 
,
(
) are given by
Here the superscript 
is omitted for the variables 
and 
in (3.7) for simplicity in notation. To understand 
and 
, we agree that they take 
when 
; while they take 
when 
. The terms 
(
;
) are expressed in terms of 
and 
(
; 
). Also 
are known functions that take value at 
, where 
when 
and 
when 
.
Since (3.7) is an algebraic linear system, the solution 
(
; 
) is uniquely solvable by 
.
Next, we give the equations and their conditions for determining the zero-order coefficient of an internal transition layer 
as follows:
We rewrite (3.8) in a different form by making the change of variables
Then, (3.8) is further written in these new variables as
From 
, it yields that the equilibrium 
of the autonomous system (3.10) is a hyperbolic saddle point. Therefore, there exists an unstable one-dimensional manifold 
. For the existence of a solution of (3.10) satisfying (3.11), we need the following assumption.
Suppose that the hyperplane 
intersects the manifold 
in the phase space 
, where 
is a parameter.
Then, 
(
) are known values after 
being solved by 
. We can get the equations and the corresponding boundary conditions to determine 
as follows:
Introducing a similar transformation as doing for (3.8), we can get
To ensure that the existence of a solution of (3.14)-(3.15), we need the following assumption.
Suppose that the hypercurve 
intersects the manifold 
in the phase space 
, where 
is a parameter.
Here it should be emphasized that under the conditions of 
and 
, the solutions 
(
) not only exist but also decay exponentially [13], or [5].
If the parameter 
is determined, 
(
) are completely known. To determine 
, it is closely related to the existence of a heteroclinic orbit connecting 
and 
in the phase space.
By the given initial values (3.13) or (3.15), we have already obtained
If we show 
, the smooth connection condition (2.11) for the zero-order is satisfied. By 
and 
, we have
Since 
(
) and 
) only depend on 
, while 
only depends on 
, the necessary condition for existence of a heteroclinic orbit connecting 
and 
at 
is given by
or
However (3.18) or (3.19) is the one to determine 
. Then, by [
], there exists an 
from (3.18) or (3.19). We can see that the process of determining 
is the one of a smooth connection. Therefore, all the zero-order terms 
have now been completely determined by the smooth connection for the zero-order coefficients of the asymptotic solution.
For the high-order terms 
(
), we have the equations and their boundary conditions as follows:
where 
represent known functions that take value at 
; 
are the known functions that only depend on those asymptotic terms whose subscript is 
; and 
and 
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 
and 
, 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 
(
) in (2.3) will be determined by an equation as follows:
Then, 
can be solved from (3.21) by 
. 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 
.
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,
where 
, 
is a parameter, such a solution 
of (4.1) and (4.2) exists by [
]–[
] [14, 15]. Then, we have 
, 
.
For the right associated problem,
where 
, 
is still a parameter, the similar reason is for the existence of 
of (4.3) and (4.4) [14, 15].
Then, we write the asymptotic expansion of 
as follows:
where 
, 
and 
.
We proceed to show that there exists an 
indeed in the neighborhood of 
such that the solution 
of the left associated problem (4.1) and (4.2) and the solution 
of the right associated problem (4.3) and (4.4) smoothly connect at 
from which we obtain the desired step-like solution.
From the asymptotic expansion of (4.5) and (4.6), we know that 
and 
are the solutions of the reduced system (2.5). In the neighborhood of 
, the boundary functions 
and 
are both exponentially small. Thus, they can be omitted in the neighborhood of 
.
We are now concerned with the equations and the boundary conditions for which 
satisfy. They can be obtained easily from (3.8) just with the replacement of 
by 
After the change of variables given by
then (4.7) can be written as
It is similar to get the equations and the boundary conditions for which 
satisfy
After the transformation given by
then (4.11) can be written as
and 
imply that there exists a first integral
of the system (4.9) that approaches 
as 
; and there exists a first integral
of the system (4.13) that approaches 
as 
.
In views of 
, from (4.15) and (4.16) we have
Then, we know from (4.2) and (4.4) that 
, 
; and 
,
for the solutions 
of the left and right associated problems. For a smooth connection of the solutions, the remaining is to prove
Let
Substituting (4.6) into (4.19), we have
where 
can be regarded as 
for simplicity.
If we take 
in (4.20), we have
Since the sign of 
is fixed, (4.21) has an opposite sign when 
is sufficiently large, for example, 
, and 
is sufficiently small. That is,
Then, there exists 
such that 
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 [
]–[
] hold. Then, there exists an 
such that there exists a step-like contrast structure solution 
(
) of the problem (2.1)-(2.2) when 
. Moreover, the following asymptotic expansion holds
Remark 4.2.
Only existence of solution with a step-like contrast structures is guarantied under the conditions of [
]–[
]. 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 
and 
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.
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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.
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Ni, M., Wang, Z. On Step-Like Contrast Structure of Singularly Perturbed Systems. Bound Value Probl 2009, 634324 (2009). https://doi.org/10.1155/2009/634324
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DOI: https://doi.org/10.1155/2009/634324