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.


Introduction
Many physical, chemical, and biological systems show periodic activity. Mathematically, they can be modeled by limit cycles of vector field. For example, in [1], 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 [2], the first two order approximate expressions of limit cycles for small positive parameter were studied by the generalized KBM method, and, in [3], the shape of the limit cycles for moderately large positive parameter was plotted by using the perturbationincremental 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 ( , ) = ∑ ∞ =0 ( , ) which satisfies the partial differential equation and the limit cycles of planar vector field ( ( , ), ( , )) are implicitly determined by ( , ) = 0. 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 [6], we know that any planar analytic system having isochronous center can be locally transformed into the above linear systeṁ= , = − by analytic variable transformation and time scale. So without losing generality, we consider analytic expression of limit cycle of perturbed planar vector fielḋ= + ( , , ), = − + ( , , ). 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 (0, 0), then ( ) is a periodic function with period 2 , where ( , ) = ( cos , sin ).

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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 [7]). If ( ) is a 2 periodic function with period 2 , then where = (1/2 ) ∫ So cos ( ) + sin ( ) ≤ + 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 [0, 2 ]. Rewrite ( ) into the following Fourier series: Integrating both sides of (6) with respect to variable from 0 to , we obtain (2) with From the integration property of uniformly convergent series, we get that ( ) is a periodic function with period 2 . If ( ) is a periodic function with period 2 , then we get 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 = , = − + (1− 2 ) up to order ( 7 ). In Section 4, we study the analytic expression of the limit cycle bifurcated from a nonlinear isochronous center.

Asymptotic Expressions of Limit Cycles
Bifurcated from the Center of = , =− Consider the following planar system: where ( , ) and ( , ) are both analytic functions, (0, 0) = (0, 0) = 0, = 1, 2, . . ., and is a small real parameter. System (9) has an isochronous center at the origin when = 0. As usual, the prime denotes derivative with respect to variable . System (9) for = 0 is called the unperturbed system, while system (9) for ̸ = 0 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.

Determination of 0 and the Poincar-Melnikov Integral.
From 0 / = 0, we get 0 ( ) ≡ 0 (constant). To determine the constant 0 in (12), the new approach we adopted is to utilize the expression of 1 ( ).
So from Lemma 2, we know 1 = 1 ( 0 ) = 0. By solving that algebraic equation, we can determine the value of constant 0 .

Remark 4.
In fact, the function 1 ( 0 ) is closely related to the first order Poincare-Melnikov integral of the perturbed system (9) near close orbit of unperturbed system (9)| =0 .
It should be pointed out that the function 1 ( 0 ) is also closely related to the first order averaging of 1-dimensional 2 -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 [11] for more details).

Determination of 1 ( ).
Substitute the value of 0 into (16); we can obtain expression of 1 ( 0 , ). Thus we obtain To determine the value of 1 , new algorithm proposed in this paper needs the expression of 2 ( ). From 2 / = 2 ( 1 , 0 , ), we get The Scientific World Journal From Lemma 2, we know . From the fact that 2 ( ) is a periodic function and Lemma 2, we get 2 ( 1 ) = 0.
By solving the above algebraic equation, we determine the value of 1 . Thus we have obtained 1 ( ) by (18).
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.

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 ( 7 ). First we make a polar coordinates transformation = cos( ) = sin( ) to system (25) and eliminate ; then we can obtain ( ) is the polar coordinates form of the limit cycles of (25) and substitute it into (26). By comparing first eight coefficients of terms , = 1, 2, . . . , 8 in both sides of the above equation, we get Here for long expressions, the formula of ( −1 , . . . , 1 , To determine the value of 1 , we compute the expression of 2 ( ): In a similar way, we determine the value of 2 and obtain following results:

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 .
Rewrite the limit cycle Γ 1 of system (38) into the following parametric form: where ( ) is given in (41). Thus from analytic transformation and time scale (37), corresponding to Γ 1 , we obtain that limit cycle of the system (36) for = (1/20) is unstable and its parametric form is the following: The shape of limit cycle of the system (25) for = 1/20 is plotted by using formula (44) in Figure 3. In Figure 3, the periodic orbit ( − 2 ) 2 + 2 = 2 of unperturbed system (36) for = 0 is drawn in sold line, and the limit cycle of the perturbed system (36) for = 1/20 is drawn in dash line.