- Research Article
- Open Access
On Step-Like Contrast Structure of Singularly Perturbed Systems
Boundary Value Problems volume 2009, Article number: 634324 (2009)
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.
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  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  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 
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  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 , 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
 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
Suppose that (2.17) is solvable with respect to and it yields a solution . That is, and .
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.
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  or . 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.
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
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  or . 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,
For the right associated problem,
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
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.
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
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  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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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.