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New idea and algorithm are proposed to compute asymptotic expression of limit cycles bifurcated from the isochronous center. Compared with known inverse integrating factor method, new algorithm to analytically computing shape of limit cycle proposed in this paper is simple and easy to apply. The applications of new algorithm to some examples are also given.

Many physical, chemical, and biological systems show periodic activity. Mathematically, they can be modeled by limit cycles of vector field. For example, in [

In 1881–1886, Poincaré defined a center of planar vector field as an isolated singular point surrounded by a family of periodic orbits. Then one interesting problem is to ask whether limit cycles appear near the periodic orbits in the vicinity of the center as the planar vector field having a center is perturbed, and what are the shapes of these limit cycles if they exist? Literatures [

If planar analytic vector field

From the periodicity of limit cycle and the property of polar coordinate system, we know that the conclusion of the lemma is true.

If

Further, if

For

If

The proof of the lemma is completed.

The main goal of this paper is to develop a new approach for computing analytically the global shape of the bifurcated limit cycles from an isochronous center and the paper is organized as follows. In Section

Consider the following planar system:

The main idea of computing asymptotic expression of limit cycles of system (

Firstly, we make a polar coordinates transformation

Secondly, to obtain the polar coordinate form of limit cycles of perturbed system (

From Lemma

Thirdly, substituting (

As to the formula of

Functions

According to (

For

For

The proof of the lemma is completed.

To obtain analytic expression of limit cycle

From

From

By noting that

From Lemma

So from Lemma

In fact, the function

In detail,

It should be pointed out that the function

Substitute the value of

To determine the value of

From Lemma

From the fact that

By solving the above algebraic equation, we determine the value of

Assuming that we have obtained the explicit expressions of

From

To determine the expression of

From Lemma

Because

By solving the algebraic equation, we obtain the value of

Thus we can compute the shape of limit cycles of system (

In this section we will apply the method just described in the above section to compute the analytic expansion of the unique limit cycle of the Van der Pol system

First we make a polar coordinates transformation

Assume

From

To determine the value of

Substitute

To determine the value of

Because

Thus explicit expression of

Substitute

So the asymptotic expansion of limit cycle of system (

The first seven terms in the above expansion of

The periodic orbit

Consider the following perturbed system:

In this subsection we start to compute the analytic expansion of the limit cycle of the perturbed system (

First let

By solving

From

By applying the algorithm described in Section

In Figure

The periodic orbit

In this subsection, we give the analytic expansion of the limit cycle of perturbed system (

Rewrite the limit cycle

Thus from analytic transformation and time scale (

The periodic orbit for

In Figure

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

The project was supported by National Natural Science Foundation of China (NSFC 11101189 and NSFC 11171135), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and Natural Science Foundation of Jiangsu Province of China (BK2012282).