International Journal of Differential Equations

Volume 2014 (2014), Article ID 430951, 6 pages

http://dx.doi.org/10.1155/2014/430951

Research Article

The Research of Periodic Solutions of Time-Varying Differential Models

Wenjun Liu,¹ Yingxin Pan,² and Zhengxin Zhou¹

¹School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China

²Department of Mathematics, Rugao Higher Normal School, Jiangsu, Rugao 226500, China

Received 20 February 2014; Accepted 2 April 2014; Published 16 April 2014

Academic Editor: Kanishka Perera

Copyright © 2014 Wenjun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We have studied the periodicity of solutions of some nonlinear time-varying differential models by using the theory of reflecting functions. We have established a new relationship between the linear differential system and the Riccati equations and applied the obtained results to discuss the behavior of periodic solutions of the Riccati equations.

1. Introduction

From the book [1] we know that a lot of biological models can be expressed by the following differential system:

which has a continuous differentiable right-hand side, and have an unique solution for their initial value problem.

Because of this nonlinear and time-varying differential system, to discuss its qualitative behavior is very difficult. Now, we use the method of reflecting function [2, 3] to study the property of solutions of (1) and got some good results.

In the present section, we introduce the concept of the reflecting function, which will be used throughout the rest of this paper.

Consider differential system

which has a continuous differentiable right-hand side, with a general solution . For each such system, the reflecting function [2, 3]is defined as . Therefore, for any solution of (2), we have.

If system (2) is-periodic with respect to, then is the Poincaré mapping of (2) over the periodThus, the solution of (2) defined onis-periodic if and only ifis a fixed point of. The stability of this periodic solution is equivalent to the stability of the fixed point.

A differentiable functionis a reflecting function of system (2) if and only if it is a solution of the Cauchy problem

If the reflecting function of (2) has the form of, then the matrixis called reflecting matrix. Thus, this matrix is a reflecting matrix of linear system, if and only if

There are many papers which are devoted to investigations of qualitative behavior of solutions of differential systems by help of reflecting functions. V. I. Mironenko and V. V. Mironenko [2–6] combined the theory of reflecting function with the integral manifold to discuss the symmetry and other geometric properties of solutions of (2), and obtained a lot of excellent new conclusions. Alisevich [7] has discussed when a linear system has triangular reflecting function. Musafirov [8] has studied when a linear system has reflecting function which can be expressed as a product of three-exponential matrix. Veresovich [9] has researched when the nonautonomous two-dimensional quadric systems are equivalent to a linear system. Maĭorovskaya [10] has established the sufficient conditions under which the quadratic systems have linear reflecting function. Zhou [11–13] has discussed the structure of reflecting function of quadratic systems and applied the obtained conclusions to study the qualitative behavior of solutions of such differential systems.

In this paper we will discuss when the system (1) has the reflecting function in the form of

where, , , and are continuously differentiable functions in. We will give the sufficient conditions for system (1) which has reflecting function of the form (5). We establish the relationship between the Riccati equation

and linear system

We apply the obtained conclusions to study the behavior of solutions of system (1) and the above Riccati equations.

In the following, we will denote, , , and so forth.

2. Main Results

Lemma 1. If the function (5) is the reflecting function of any system, then for small, it can be written in the form, in which

Proof. As the function (5) is a reflecting function, so , ; that is,

Equating the coefficients of the same power of, we get

The relation (11) implies that

Thus,

Substituting it into (10) and (12), we get

So

The proof is finished.

By Lemma 1, in the following, we all suppose, in which.

Theorem 2. Suppose that , , , and are the solutions of the Cauchy problem

whereand , , .

Then

is the reflecting function of system (1). Therefore, nearby , , where.

Proof. Putting

then by (18), we have

By the uniqueness of the solution of linear partial differential equations, we get

that is,

Obviously,is the first integral of system (18); by the initial conditions, we haveThus, the functions andhave not the common zeros. As, so in the nearby area of, .

Using the conditions of the present theorem, it is not difficult to check that function (20) satisfies the relation (3) with respect to So, the function (20) is the reflecting function of (1). The proof is completed.

Corollary 3. If all the conditions of Theorem 2 are satisfied, and the system (1) is-periodic with respect to, then its Poincaré mapping can be expressed by

So, the solutionof (1) defined on-periodic, if and only ifis the solution of equations

Corollary 4. Suppose that

andis continuously differentiable and satisfies

Then is the reflecting function of system (1). In addition, if the system (1) is a-periodic system with respect to, then all the solutions of (1) defined onare-periodic.

Proof. It is not difficult to check that, under the conditions of above corollary, the functionis the solution of the Cauchy problem (3), so it is a reflecting function of (1). In view of, soand. Thus, the conclusion of above corollary is true.

Example 5. Differential system

has reflecting function. Where , When , , all the solutions of above system defined onare-periodic.

Example 6. Differential system

has reflecting function . Where , , .

If the functions , , are-periodic with respect to , then all the solutions of above system defined on are -periodic.

Similar to Corollary 4, we get the following.

Corollary 7. If

is continuously differentiable and satisfies

Then is the reflecting function of system (1). In addition, if the system (1) is a-periodic system, then all the solutions of (1) defined on are-periodic.

Example 8. Differential system

has reflecting function .

Where, , . Besides, if functions , , are-periodic with respect to , then all the solutions of above system defined onare-periodic.

Corollary 9. Suppose that

and then is the reflecting function of (1). In addition, if the system (1) is a -periodic system, then the solution of (1) defined on is-periodic, and if and only if , is a natural number.

Proof. In Theorem 2 taking , , , under the above hypothesis, the function is the reflecting function of (1). Thus, when the system (1) is -periodic, its Poincaré mapping can be expressed by, its solution defined on is-periodic, and if and only if , it implies the conclusion of the present corollary.

Example 10. Differential system

has reflecting function

and all the solutions of above system defined onare-periodic, where functions and are arbitrary continuously differentiable odd and-periodic with respect to .

Theorem 11. Let the reflecting function of linear system

have the form

Then the function

is the reflecting function of the Riccati equation

Proof. Asis the reflecting matrix of the linear system (37), thus,

and it implies

where , , , and . Similar to Theorem 2, we get function (39) is the reflecting function of the Riccati equation (40).

Corollary 12. If and the conditions of Theorem 11 are satisfied, then

(1)if and , then the Riccati (40) has two-periodic solutions. Ifand, (40) has no one-periodic solution. Ifand, (40) has a unique-periodic solution;(2)ifand, then (40) has a unique-periodic solution. If and, , all the solutions of (40) defined onare-periodic; Ifand, , (40) has no one-periodic solution.

Proof. As (39) is the reflecting function of (40) and . Thus, the Poincaré mapping can be expressed by , and the solutions of (40) defined on are-periodic, if and only if is a solution of equation: ,

and from this relation and, it is easy to deduce the present conclusion.

Theorem 13. Suppose that

(1), are continuous and;(2)is continuously differentiable and and satisfies

Then the function (39) is the reflecting function of the Riccati equation (40). In addition, if, then all the solutions of (40) defined onare-periodic, where

Proof. It is not difficult to check that under the hypothesis of Theorem 13 the functions , , are the solution of the Cauchy problem (42), so, the function (39) is-periodic reflecting function of (40); by [1] it implies the result of the present theorem.

From a similar discussion we can get the following results.

Theorem 14. Suppose that

(1), are continuous and ;(2)is continuously differentiable and and satisfies

Then the function (39) is the reflecting function of the Riccati equation (40). In addition, if , then all the solutions of (40) defined onare-periodic, where

Theorem 15. If there is a continuously differentiable even functionand

(1) and are continuous and ;(2) is continuously differentiable and and satisfies

Then the function (39) is the reflecting function of the Riccati equation (40). In addition, if , , then all the solutions of (40) defined onare-periodic, where

Example 16. The Riccati equation

has reflecting function

So, all the solutions of the equation above defined onare-periodic.

Example 17. The Riccati equation

has reflecting function

where and . And all the solutions of the equation above defined onare-periodic.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the NSF of Jiangsu of China under BK2012682, 201311117084X, and the NSF of China under Grant 11271026, 61374010.

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