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.

As we know [

Since the closed orbits of (

To discuss the center-focus problem, there are Lyapunov’s method and the others; see the works of Z. Zhang and so forth [

In this paper, we apply the method of Mironenko [

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

If system (

A differentiable function

If

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.

Now, let us consider differential equation (

Suppose that

If all the solutions of (

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

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.

Consider system

Now, we use the method of Lyapunov to discuss is the

Equating the coefficients of like powers of

From above, we see that the calculation of the expression of

Now, we try to use the method of Mironenko to discuss is

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

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

If

By the assumptions, we see that

If there is a function

Let

In fact, by relation (

If there is a differentiable

This result is implied by Theorem

Taking

For (

By the assumptions, it is not difficult to check that

For (

By condition (

Suppose that for (

By the assumptions and [

Theorem

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

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