Abstract and Applied Analysis

Volume 2014 (2014), Article ID 926538, 5 pages

http://dx.doi.org/10.1155/2014/926538

Research Article

A New Method for Research on the Center-Focus Problem of Differential Systems

Zhengxin Zhou

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

Received 13 December 2013; Accepted 17 February 2014; Published 20 March 2014

Academic Editor: Douglas R. Anderson

Copyright © 2014 Zhengxin Zhou. 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 will introduce Mironenko’s method to discuss the Poincaré center-focus problem, and compare the methods of Lyapunov and Mironenko. We apply the Mironenko method to discuss the qualitative behavior of solutions of some planar polynomial differential systems and derive the sufficient conditions for a critical point to be a center.

1. Introduction

As we know [1, 2], for polynomial differential system

where and are real constants, there has been a longstanding problem, called the Poincaré center-focus problem; for the system (1) find explicit conditions of and under which (1) has a center at the origin ; that is, all the orbits nearby are closed. The problem is equivalent to an analogue for a corresponding periodic equation

to have periodic solutions. To see this let us note that the phase curves of (1) near the origin in polar coordinates , are determined by (2), where and are polynomials in and .

Since the closed orbits of (1) correspond to -periodic solutions of (2), the planar vector field (1) has a center at (0, 0) if and only if (2) has a center at ; that is, all the solutions nearby are periodic: [1, 2].

To discuss the center-focus problem, there are Lyapunov’s method and the others; see the works of Z. Zhang and so forth [2–8]. One of the most commonly used methods is Lyapunov’s method. However, sometimes, the use of this method to calculate the focus quantities is very difficult and the amount of calculation is very large.

In this paper, we apply the method of Mironenko [9] (reflecting function method) to study the qualitative behavior of solutions of (2) and give the sufficient conditions for to be a center. We give a few examples to illustrate that, sometimes, Mironenko's method is better than Lyapunov's method.

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 continuously differentiable right-hand side and with a general solution . For each such system, the reflecting function is defined (see Mironenko's [9]) as . Therefore, for any solution of (3), we have , .

If system (3) is -periodic with respect to , and is its reflecting function, then is the Poincaré mapping of (3) over the period . Thus, the solution of (3) defined on is -periodic if and only if is a fixed point of . The stability of this periodic solution is equivalent to the stability of the fixed point .

A differentiable function is a reflecting function of system (3) if and only if it is a solution of the Cauchy problem

If is a reflecting function of system (3), then it is also reflecting function of the following systems:

where is an arbitrary vector function. Thus, the shift operators [9, 10] on interval for such systems coincide, and the qualitative behaviors of the periodic solutions of these systems are the same. See papers [11–18] in which reflecting functions are also applied to study the qualitative behavior of solutions of some nonlinear and nonautonomous differential systems.

In the following, we always assume that all equations in this paper have a continuously differentiable right-hand side and have a unique solution for their initial value problem.

2. Main Results

Now, let us consider differential equation (2).

Lemma 1 ([see [9]). Suppose that is the reflecting function of (2) and . Then all the solutions of (2) defined on are -periodic; that is, is a center.

Lemma 2 ([see [9]). If all the solutions of (2) are -periodic, then its reflecting function is -periodic, too.

First, we introduce the Mironenko’s method. Suppose that

is the reflecting function of (2), where . Substituting it into

and equating the coefficients of like powers of , we get the following infinite equations

If , then and is a center. If this is a and , then , where is a constant and when , is an unstable focus; when , is a stable focus.

By the above description, it seems that Mironenko's method is similar to Lyapunov's method, but do not forget that reflecting function has some good symmetry properties, which make our calculations greatly reduced. The following example will illustrate this advantage.

Example 3. Consider system

Taking , it is transformed to equation

Now, we use the method of Lyapunov to discuss is the a center or focus? Suppose that

is a solution of (10). Substituting it into (10), we get

where .

Equating the coefficients of like powers of implies

Solving it we get

It implies

so

From above, we see that the calculation of the expression of will become more difficult. Thus, it is not easy to judge is a center?

Now, we try to use the method of Mironenko to discuss is a center? In (7) taking

we get

Equating the coefficients of like powers of implies

and so forth. Now we can conjecture that ; that is, . It is not difficult to check that is a solution of (7), so it is the reflecting function of (10). As , so the origin point of system (9) is a center.

Such example shows us that, sometimes, the method of Mironenko is better than Lyapunov’s method. Just this advantage is not enough. By the equivalence, we know that also is the reflecting function of the following equations:

where is an arbitrary function. If we take

where is a positive integer, then (20) becomes

Taking , then (22) is transformed to

and its origin point is a center too.

Thus, by using the method of Mironenko we not only solve a center-focus problem, but at the same time open a class of differential equations with the same character of point . Therefore, we can say, sometimes, the method of Mironenko is more effective than Lyapunov’s method.

Theorem 4. If is a solution of equation such that and satisfies (7), then all the solutions of (2) determined on are -periodic; that is, is a center.

Proof. By the assumptions, we see that is a reflecting function of (2) and -periodic; thus by Lemma 1, the present conclusion is correct.

Theorem 5. If there is a function , which satisfies

where . Then all the solutions of (2) determined on are -periodic; that is, is a center.

Proof. Let be the reflecting function of (2). Now, we prove that it satisfies .

In fact, by relation (24), for any solution of (2), we have

In view of , we have [9, page 65]

that is,

which implies that is a -periodic function with respect to . By Lemma 1, the proof of the present theorem is finished.

Corollary 6. If there is a differentiable -periodic function such that

are -periodic and odd functions, then all the solutions defined on of equation

are -periodic.

This result is implied by Theorem 5 if we take and put

Example 7. Taking , we get that all the solutions determined on of the Riccati equation

are -periodic.

Theorem 8. For (2), suppose that

, , . Then is a center.

Proof. By the assumptions, it is not difficult to check that is the reflecting function of (2), so is a center.

Theorem 9. For (2), suppose that and

where , , . Then all the solutions of (2) and the following equations

determined on are -periodic; that is, is a center, where is a continuously differentiable -periodic function with respect to and .

Proof. By condition (33), we can check that is the reflecting function of (2). As , so and are -periodic functions with respect to . Then by Lemma 1, is a center.

Theorem 10. Suppose that for (2) is a center and functions are -periodic with respect to and satisfy

Then for the following equations

is a center, too. Here are arbitrary continuously differentiable -periodic odd functions.

Proof. By the assumptions and [9, 14], we know that the reflecting functions of (2) and (36) coincide and all the solutions of (2) determined on are -periodic. By Lemma 2, we know that the common reflecting function is -periodic; thus all the solutions of (36) determined on are -periodic; that is, is a center, too.

Remark 11. Theorem 10 shows us that if the center-focus problem for (2) is already solved, then the same problem for (36) is solved, too.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work supported by the NSF of Jiangsu of China under Grant no. BK2012682 and the NSF of China under Grant nos. 11271026, 61374010.

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