Abstract
We investigate the step-type contrast structure for high dimensional Tikhonov system with Neumann boundary conditions. We not only propose a key condition with the existence of the number of mutually independent first integrals under which there exists a step-type contrast structure, but also determine where an internal transition time is. Using the method of boundary function, we construct the formal asymptotic solution and give the analytical expression for the higher order terms. At the same time, the uniformly valid asymptotic expansion and the existence of such an available step-type contrast structure are obtained by sewing connection method.
1. Introduction
The fundamental characteristic of contrast structure is that there is (or multiple ) within the domain of interest, which is called an internal transition point. The position of which is unknown in advance needs to be determined thereafter. In the neighborhood of , the genuine solution will have an abrupt structure change and in the different side of , it will approach different reduced solutions when the small parameter . The contrast structure in singularly perturbed problems is classified as a step-type contrast structure [1–6] or a spike-type contrast structure [7–9]. In the West, the study on this issue is mainly by the method of dynamic systems or geometric method [10–12]. In recent years, the study on contrast structures is still a hot but difficult research topic in the theory of singularly perturbed problem, especially for high dimensional singularly perturbed system [13]. In fact, the existence of a step-type contrast structure is closely related to the existence of a heteroclinic orbit of its auxiliary system in its corresponding phase space. However, how to find and construct such an orbit in a high dimensional dynamic is itself difficult in general in the theory of qualitative analysis. This is why it is nontrivial in extending to high dimensional case from the contrast structures in plane. Using the method of boundary function and sewing connection, Ni and Wang [14] investigate the step-type contrast structure for the following singular perturbed system:where , are -dimensional vectors. In this paper, we extend the existence of step-type contrast structure to the following:where is a small parameter, , are -dimensional vectors, , are -dimensional vectors, , are matrixes, is an unit matrix, and is an unit matrix. , are matrixes, is an unit matrix, and is an unit matrix.
2. Assumptions
The following assumptions are fundamental in theory for the problem in question.
() Suppose that and are sufficiently smooth on the domain , where , are real numbers.
The reduced system of (2) is given by
() Suppose that the reduced subsystem of (2) given by has two isolated solutions on and , where , .
() For , suppose that and have characteristic roots and , , respectively, which satisfy where .
() shows that, for fixed , the auxiliary system given byhas two equilibriums and which are all hyperbolic saddle points.
3. Construction of the Asymptotic Expansion
Let be the transition point and it is given by with which will be given in the following.
Suppose that the asymptotic solution is composed of two parts.
The left problem (:
The right problem (:where are parameters which will be determined in the following but related to . For convenience, let .
To obtain the step-type solution, we need
Let . Suppose the formal asymptotic solutions for the left and the right problem arerespectively, where , , and . are coefficients of regular terms; are coefficients of the boundary layer terms at ; are coefficients of the boundary layer terms at ; are the left and the right coefficients of internal transition terms at . Furthermore, , , and .
3.1. Construction of the Zero-Order Terms
Substituting (15), (16) into (8)–(10) and (11)–(13), respectively, by the boundary function method [15], we have By () and the assumed solution type, we know while satisfy the following problems:respectively.
() Suppose the solution of (19), (20) and the solution of (21), (22) are transversal at , where
Next, we give the equations and their conditions for determining the zero-order coefficient of as follows:Let , . Then we haveObviously, is a hyperbolic saddle point of (29) and is a hyperbolic saddle point of (30).
The existence of the solutions for (23)–(28) will be given in the following. If they are used in the following calculation process, we think that they are known. Obviously, they are associated with . Moreover, by the boundary conditions , and , , we have , .
3.2. Construction of the Higher Order Terms
For , we have the equations and their boundary conditions as follows:where , take value at and , respectively. , , and have the same significance. , are known functions and are determined.
satisfy the following problems:where . , are determined functions.
By (38) and , we know that Then, . So we have
The equations to determine are given bywhere , , while , are known functions.
By (42) and , we know that Then, . So we have
And then can be obtained.
() Suppose that the solution of (31), (32) and the solution of (34), (35) are transversal at .
The equations to determine are given byHere While has the same significance. , , and are determined, excluding .
By and (46) we have Substituting them into (45) we obtain under the initial condition (47). Moreover, are related to .
As for , it satisfies the following boundary value problem:Here While has the same significance. , , and are determined, excluding .
By and (45) we have Substituting it into (28), we get a first-order linear equation, so exists under the initial condition (47). And then exists. Substituting into (46) we obtain under the initial condition (46). Moreover, are related to .
In the following, we will give the analytical expression for . Because are solved, we rewrite (45), (50) as follows: where Writing it into block structure given by we obtain where , , while and are the solutions of respectively. Consider where , , while and are solutions of respectively. The system to determine , is similar to the system to determine , so we can obtain and using the same method.
4. The Existence of the Heteroclinic Orbit
We consider the associated systemfor (25), (26), which coincide with the auxiliary system (6) when . Obviously, there are two equilibriums and for (64). Whether the step-type solution from to for problem (2), (3) exists or not largely depends on the existence of the heteroclinic orbit for system (64) which connects to .
The following assumptions ensure the existence of such heteroclinic orbit.
() For fixed , suppose that there exist linearly independent first integrals for system (64) given bywhere are independent arbitrary parameters.
Then, the orbit passing through is given byThe orbit passing through is given by
It is noted thatIf (68) hold, a heteroclinic orbit connecting and can be obtained. Meanwhile, will be determined through (68) under the following assumption.
() Suppose that (68) are compatible and have a solution
Under condition (), there exists a heteroclinic orbit which connects and .
Let , . are dimensional unstable manifolds which pass and are dimensional stable manifolds which pass . Then, . By (66), we know . So . Similarly, , while by (67). So .
Take , and then . Because of the existence of the heteroclinic orbit, we have In (69), the coefficient matrix for is as follows:
() Suppose that
According to (), we know that the solutions for the left and the right problem satisfy condition (14). So the federated system for (50) has a solution which satisfies , , where By [16], system (71) has an exponential dichotomy in and . is Fredholm with index zero [17]. Because , there exist which satisfy If the solution for (71) exists, the necessary and sufficient condition is given by We rewrite (73) as follows:
() Suppose that .
Under this assumption, is completely determined. So far, we have already determined all coefficients of the formal asymptotic solution.
5. Existence of a Step-Type Solution
The formal asymptotic solution of is given byand the formal asymptotic solution of is given by
Substituting (75), (76) into the left problem and the right problem, respectively, completely similar to the previous calculation process, we can obtain the systems to determine all the coefficients of the asymptotic expansions. It is noted that is given by while Obviously, the system to determine is the same as the front. When , we only need to change into in (47), (52).
Let where . Since and at are both exponentially small quantities, without loss of generality, we can regard as follows: In terms of the boundary value conditions of the left and right associated problems, we have . Then (80) yields In the same time, When takes its value from to , (82) can be written aswhere and are the first components and the rest of components of respectively. and are the first components and the rest of the components of respectively.
By (), there exist , such that (83) is equal to zero. Namely, (82) is equal to zero. Then, we obtain a step-type contrast structure at the neighborhood of . Similarly, we can prove . In summary, we have the following result.
Theorem 1. Suppose that hold. Problem (2), (3) has a step-type contrast structure solution . Moreover, the following asymptotic expansion holds:
Conflict of Interests
Authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors acknowledge the following: (1) the National Natural Science Funds (no. 11501236); (2) the National Natural Science Funds (no. 11471118); (3) the National Natural Science Funds (nos. 30921064, 90820307) supported by Knowledge Innovation Project in the Chinese Academy; (4) Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China.