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On the Shape of Limit Cycles That Bifurcate from Isochronous Center
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 , Van der Pol proved that a closed trajectory of a self-sustained oscillation occurring in a vacuum tube circuit was a limit cycle as defined by Poincaré. The study of limit cycles of real general planar vector field is closely related to Hilbert’s 16th Problem. As to the strongly nonlinear oscillation equation , , in , the first two order approximate expressions of limit cycles for small positive parameter were studied by the generalized KBM method, and, in , the shape of the limit cycles for moderately large positive parameter was plotted by using the perturbation-incremental method.
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 [4, 5] have applied the method of inverse integrating factor to analytically compute global shape of the limit cycles bifurcated from analytic isochronous center. The main idea of determining the shape of limit cycles of planar vector field in [4, 5] is to determine function which satisfies the partial differential equation and the limit cycles of planar vector field are implicitly determined by . In other words, if one tries to find analytic expression of limit cycle, one should solve linear partial differential equations recursively. In this paper, a new idea and algorithm are developed to analytically compute the shape of the limit cycles bifurcated from the isochronous center. From Theorem 3.2 in , we know that any planar analytic system having isochronous center can be locally transformed into the above linear system , by analytic variable transformation and time scale. So without losing generality, we consider analytic expression of limit cycle of perturbed planar vector field , . The new algorithm proposed in the paper is based on the following lemmas.
Lemma 1. If planar analytic vector field has a limit cycle surrounding the origin , then is a periodic function with period , where .
Proof. From the periodicity of limit cycle and the property of polar coordinate system, we know that the conclusion of the lemma is true.
Lemma 2 (see ). If is a periodic function with period , then
where and is a periodic function with period .
Further, if is periodic function, then .
Proof. For is a periodic function with period , so Fourier coefficients of functions and have the following relations:
By applying Bessel inequality, we get
By applying comparison test for convergence of series of functions, we get that Fourier series of is uniformly convergent to on . Rewrite into the following Fourier series:
Integrating both sides of (6) with respect to variable from to , we obtain (2) with
From the integration property of uniformly convergent series, we get that is a periodic function with period .
If is a periodic function with period , then we get . From , we conclude that .
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 2, we develop a new algorithm to compute analytic expansion, up to arbitrary order of the parameter , of the limit cycles bifurcated from linear isochronous center. As applications, in Section 3, we compute the analytic expression of the unique limit cycle of the Van der Pol system , up to order . In Section 4, we study the analytic expression of the limit cycle bifurcated from a nonlinear isochronous center.
2. Asymptotic Expressions of Limit Cycles Bifurcated from the Center of ,
Consider the following planar system: where and are both analytic functions, , , and is a small real parameter. System (9) has an isochronous center at the origin when . As usual, the prime denotes derivative with respect to variable . System (9) for is called the unperturbed system, while system (9) for is called the perturbed one. Then the problem of studying shape of limit cycles bifurcated from isochronous center is to determine the number and analytic expansions of the families of limit cycles which emerge from the periodic orbits of the unperturbed system as the parameter is varied.
The main idea of computing asymptotic expression of limit cycles of system (9) is the following.
Firstly, we make a polar coordinates transformation , to system (9). By eliminating the variable , we obtain By noticing that , are both analytic functions, we rewrite system (10) into the following form: where are analytic functions about , , .
As to the formula of , we have the following lemma.
Lemma 3. Functions obtained in (13) have the following properties: where is analytic function about , .
Proof. According to (11) and (12), we get that
For , it is easy to get that .
For , the term on the right hand side of (15) contributing to is . In detail, the constant in the term determines . Noticing that , we get that only contains the term . In other words, if function contains variable , then appearing in the terms in the right hand side of (15) at least have term . This is contradiction, for corresponds to term in the right hand side of (15). By using similar analysis, it can be shown that cannot contain term . Therefore, , .
The proof of the lemma is completed.
2.1. Determination of and the Poincaré-Melnikov Integral
From , we get . To determine the constant in (12), the new approach we adopted is to utilize the expression of .
From , we obtain
By noting that is a periodic function, according to Lemma 2, we know where , and is periodic function with period .
So from Lemma 2, we know . By solving that algebraic equation, we can determine the value of constant .
Remark 4. In fact, the function is closely related to the first order Poincare-Melnikov integral of the perturbed system (9) near close orbit of unperturbed system (9).
In detail, , where close orbit . From [8, 9], we know is the first order Poincare-Melnikov integral. So the zeros of are closely related to the number and position of limit cycles of the perturbed system (9).
It should be pointed out that the function is also closely related to the first order averaging of 1-dimensional -periodic differential equation. First order (resp., second order) averaging method to study the existence and number of periodic orbits of planar differential equation is proposed in [10, 11]. The approach of high order averaging method is based on Brouwer degree theory (see  for more details).
2.2. Determination of
Substitute the value of into (16); we can obtain expression of . Thus we obtain
To determine the value of , new algorithm proposed in this paper needs the expression of . From , we get
From Lemma 2, we know where .
From the fact that is a periodic function and Lemma 2, we get .
By solving the above algebraic equation, we determine the value of . Thus we have obtained by (18).
2.3. Determination of
Assuming that we have obtained the explicit expressions of , now we start to determine .
From , we get
To determine the expression of is to determine the value of . According to the algorithm proposed in this paper, we resort to the expression of . From , we get
From Lemma 2, we know where
Because is a periodic function, from Lemma 2, we get that .
By solving the algebraic equation, we obtain the value of ; thus we determine by (20).
Thus we can compute the shape of limit cycles of system (9) to any given order of explicitly and recursively.
3. The Shape of Limit Cycle of Van der Pol 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 up to .
First we make a polar coordinates transformation to system (25) and eliminate ; then we can obtain
Assume is the polar coordinates form of the limit cycles of (25) and substitute it into (26). By comparing first eight coefficients of terms in both sides of the above equation, we get Here for long expressions, the formula of , is omitted.
From , we get that is arbitrary constant.
To determine the value of , we compute the following expression of where Because , so we get .
Substitute into (28); we obtain
To determine the value of , we compute the expression of : where
Because , so we get .
Thus explicit expression of is given by (30) with .
Substitute and into (31); we obtain In a similar way, we determine the value of and obtain following results:
So the asymptotic expansion of limit cycle of system (25) for and small is the following:
The first seven terms in the above expansion of are similar to the ones given in Section 3 of . Here we present the expression of obtained in our method which was omitted in  for its long expression. By applying expansion (35), the shapes of limit cycles of Van der Pol system (25) for the values of , , , are plotted in Figure 1. The periodic orbit of system (25) for is drawn in solid line, the limit cycle of system (25) for is drawn in dashed line, the limit cycle of system (25) for is drawn in solid line, and the limit cycle of system (25) for is drawn in dotted line.
4. The Shape of Limit Cycle of Perturbations of a System Having Nonlinear Isochronous Center
Consider the following perturbed system: From , we know that as , nonlinear system (36) has isochronous center . To utilize new algorithm introduced in Section 2 to study the number and shape of limit cycles of perturbed system (36), we first apply the following analytic variable transformation: to system (36) and get
4.1. The Shape of Limit Cycle of the Perturbed System (38)
In this subsection we start to compute the analytic expansion of the limit cycle of the perturbed system (38) to the second order of .
By solving , we get the positive solution .
In Figure 2 we illustrate the shape of the limit cycle of the system (38) by using formula (41) for the values . The periodic orbit of system (38) for is drawn in solid line, and the limit cycle is drawn in dash line. We have also plotted the limit cycle for the value by using the Runge-Kutta method in Figure 2. The close curve obtained numerically coincides with the one obtained analytically and we cannot distinguish between them with the eyes.
4.2. The Shape of Limit Cycle of the Original Perturbed System (36)
In this subsection, we give the analytic expansion of the limit cycle of perturbed system (36) to the second order of .
Thus from analytic transformation and time scale (37), corresponding to , we obtain that limit cycle of the system (36) for is unstable and its parametric form is the following: The shape of limit cycle of the system (25) for is plotted by using formula (44) in Figure 3.
Conflict of Interests
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).
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Copyright © 2014 Guang Chen and Yuhai Wu. 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.