TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 320406 10.1155/2014/320406 320406 Research Article On the Shape of Limit Cycles That Bifurcate from Isochronous Center Chen Guang Wu Yuhai Han M. Jin Z. Xia Y. 1 Faculty of Science Jiangsu University Zhenjiang, Jiangsu 212013 China ujs.edu.cn 2014 1932014 2014 07 12 2013 10 02 2014 19 3 2014 2014 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.

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.

1. Introduction

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 d x / d t = y , d y / d t = g ( x ) + ɛ    f ( x , y ) , in , the first two order approximate expressions of limit cycles for small positive parameter ɛ were studied by the generalized KBM method, and, in , the shape of the limit cycles for moderately large positive parameter ɛ 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 ( P ( x , y ) , Q ( x , y ) ) in [4, 5] is to determine function V ( x , y ) = k = 0 ε k V k ( x , y ) which satisfies the partial differential equation (1) P V x + Q V y = ( P x + Q y ) V , and the limit cycles of planar vector field ( P ( x , y ) , Q ( x , y ) ) are implicitly determined by V ( x , y ) = 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 , we know that any planar analytic system having isochronous center can be locally transformed into the above linear system x ˙ = y , y ˙ = - x by analytic variable transformation and time scale. So without losing generality, we consider analytic expression of limit cycle of perturbed planar vector field x ˙ = y + ɛ U ( x , y , ɛ ) , y ˙ = - x + ɛ V ( x , y , ɛ ) . The new algorithm proposed in the paper is based on the following lemmas.

Lemma 1.

If planar analytic vector field ( P ( x , y ) , Q ( x , y ) ) has a limit cycle r = r ( θ ) surrounding the origin O ( 0,0 ) , then r ( θ ) is a periodic function with period 2 π , where ( x , y ) = ( r cos θ , r sin θ ) .

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 [<xref ref-type="bibr" rid="B7">7</xref>]).

If f ( θ ) is a C 2 periodic function with period 2 π , then (2) F ( θ ) = 0 θ f ( ξ ) d ξ = g θ + φ ( θ ) , where g = ( 1 / 2 π ) 0 2 π f ( ξ ) d ξ and φ ( θ ) is a periodic function with period 2 π .

Further, if F ( θ ) is periodic function, then g = 0 .

Proof.

For f ( ξ ) is a C 2 periodic function with period 2 π , so Fourier coefficients of functions f ( ξ ) and f ( ξ ) have the following relations: (3) a 0 = 1 π 0 2 π f ( ξ ) d ξ = 1 π [ f ( 2 π ) - f ( 0 ) ] = 0 , a n = 1 π 0 2 π f ( ξ ) cos ( n ξ ) d ξ = 1 π f ( ξ ) cos ( n ξ ) | 0 2 π + n π 0 2 π f ( ξ ) sin ( n ξ ) d ξ = n b n , n = 1,2 , , b n = 1 π 0 2 π f ( ξ ) sin ( n ξ ) d ξ = 1 π f ( ξ ) sin ( n ξ ) | 0 2 π - n π 0 2 π f ( ξ ) cos ( n ξ ) d ξ = - n a n , n = 1,2 , . So (4) | a n cos ( n θ ) + b n sin ( n θ ) | | a n | + | b n | = | a n | n + | b n | n 1 2 ( a n 2 + b n 2 ) + 1 n 2 . By applying Bessel inequality, we get (5) n = 1 ( a n 2 + b n 2 ) 1 π 0 2 π f 2 ( ξ ) d ξ . By applying comparison test for convergence of series of functions, we get that Fourier series of f ( ξ ) is uniformly convergent to f ( ξ ) on [ 0,2 π ] . Rewrite f ( θ ) into the following Fourier series: (6) f ( θ ) = a 0 2 + n = 1 ( a n cos ( n θ ) + b n sin ( n θ ) ) , where (7) a 0 2 = 1 2 π 0 2 π f ( ξ ) d ξ = g . Integrating both sides of (6) with respect to variable θ from 0 to θ , we obtain (2) with (8) φ ( θ ) = n = 1 [ a n n sin ( n θ ) - b n n ( cos ( n θ ) - 1 ) ] . From the integration property of uniformly convergent series, we get that φ ( θ ) is a periodic function with period 2 π .

If F ( θ ) is a periodic function with period 2 π , then we get g · ( 2 π - 0 ) + φ ( 2 π ) - φ ( 0 ) = 0 . From φ ( 2 π ) = φ ( 0 ) , we conclude that g = 0 .

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 ɛ , 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 x = y , y = - x + ɛ ( 1 - x 2 ) y up to order o ( ε 7 ) . 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 <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M58"> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi></mml:mrow> <mml:mrow> <mml:mi>′</mml:mi></mml:mrow> </mml:msup> <mml:mo mathvariant="bold">=</mml:mo> <mml:mi>y</mml:mi></mml:math> </inline-formula>, <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M59"> <mml:msup> <mml:mrow> <mml:mi>y</mml:mi></mml:mrow> <mml:mrow> <mml:mi>′</mml:mi></mml:mrow> </mml:msup> <mml:mo mathvariant="bold">=</mml:mo> <mml:mo mathvariant="bold">-</mml:mo> <mml:mi>x</mml:mi></mml:math> </inline-formula>

Consider the following planar system: (9) x = y + k = 1 ε k p k ( x , y ) y + ɛ U ( x , y , ɛ ) , y = - x + k = 1 ε k q k ( x , y ) - x + ɛ V ( x , y , ɛ ) , where p k ( x , y ) and  q k ( x , y ) are both analytic functions, p k ( 0,0 ) = q k ( 0,0 ) = 0 , k = 1,2 , , and ɛ is a small real parameter. System (9) has an isochronous center at the origin when ɛ = 0 . As usual, the prime denotes derivative with respect to variable t . System (9) for ɛ = 0 is called the unperturbed system, while system (9) for ɛ 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 ɛ 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 x = r cos θ , y = r sin θ to system (9). By eliminating the variable t , we obtain (10) d r d θ = ɛ ( U cos θ + V sin θ ) - 1 + ɛ ( V cos θ - U sin θ ) / r . By noticing that p k ( x , y ) , q k ( x , y ) are both analytic functions, we rewrite system (10) into the following form: (11) d r d θ = R 1 ( r , θ ) ɛ + R 2 ( r , θ ) ε 2 + R 3 ( r , θ ) ε 3 + , where R 1 ( r , θ ) , R 2 ( r , θ ) , are analytic functions about r , cos ( θ ) , sin ( θ ) .

Secondly, to obtain the polar coordinate form of limit cycles of perturbed system (9), we look for the following analytic solution to (11) as ɛ 0 , (12) r ( θ ) = k = 0 ε k r k ( θ ) .

From Lemma 1, we know that if r ( θ ) is limit cycle of system (10), then from the periodicity of r ( θ ) , we get that r k ( θ ) is periodic function with period 2 π , where k = 0,1 , 2 , .

Thirdly, substituting (12) into (11) and considering the k th order terms of ɛ in the obtained system, then we can obtain a series of equations: (13) d r k d θ = f ~ k ( θ , r 0 , r 1 , r 2 , ) ,    k = 0,1 , 2 , .

As to the formula of f ~ k ( θ , r 0 , r 1 , r 2 , ) , we have the following lemma.

Lemma 3.

Functions f ~ k ( θ , r 0 , r 1 , r 2 , ) obtained in (13) have the following properties: (14) f ~ 0 ( θ , r 0 , r 1 , r 2 , ) = 0 , f ~ 1 ( θ , r 0 , r 1 , r 2 , ) = f 1 ( r 0 , θ ) , f ~ 2 ( θ , r 0 , r 1 , r 2 , ) = f 2 ( r 1 , r 0 , θ ) , f ~ k ( θ , r 0 , r 1 , r 2 , ) = f k ( r k - 1 , r k - 2 , , r 1 , r 0 , θ ) , where f k ( r k - 1 , r k - 2 , , r 1 , r 0 , θ ) is analytic function about r k - 1 , r k - 2 , , r 1 , r 0 , cos ( θ ) , sin ( θ ) , k = 0,1 , 2 , .

Proof.

According to (11) and (12), we get that (15) k = 0 ε k d r k ( θ ) d θ = R 1 ( r , θ ) ɛ + R 2 ( r , θ ) ε 2 + R 3 ( r , θ ) ε 3 + .

For k = 0 , it is easy to get that f ~ 0 ( θ , r 0 , r 1 , r 2 , ) = 0 .

For k = 1 , the term on the right hand side of (15) contributing to ε 1 is ɛ R 1 ( r , θ ) . In detail, the constant in the term R 1 ( r , θ ) determines f ~ 1 ( θ , r 0 , r 1 , r 2 , ) . Noticing that r ( θ ) = r 0 + ɛ r 1 ( θ ) + ε 2 r 2 ( θ ) + , we get that f ~ 1 ( θ , r 0 , r 1 , r 2 , ) only contains the term r 0 . In other words, if function f ~ 1 ( θ , r 0 , r 1 , r 2 , ) contains variable r n , n 1 , then r n appearing in the terms in the right hand side of (15) at least have term ε n + 1 . This is contradiction, for f ~ 1 ( θ , r 0 , r 1 , r 2 , ) corresponds to ε 1 term in the right hand side of (15). By using similar analysis, it can be shown that f ~ n ( θ , r 0 , r 1 , r 2 , ) cannot contain term r i , i n . Therefore, f ~ k ( θ , r 0 , r 1 , r 2 , ) = f k ( r k - 1 , r k - 2 , , r 1 , r 0 , θ ) , k = 0,1 , 2 , .

The proof of the lemma is completed.

To obtain analytic expression of limit cycle r = r ( θ ) of system (9), we need to determine r k ( θ ) in (12), k = 0,1 , 2 , . From Lemma 3, we know that the determinations of r k ( θ ) in (12) are recursive.

2.1. Determination of <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M123"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>r</mml:mi></mml:mrow> <mml:mrow> <mml:mn>0</mml:mn></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula> and the Poincaré-Melnikov Integral

From d r 0 / d θ = 0 , we get r 0 ( θ ) r 0 ( constant ) . To determine the constant r 0 in (12), the new approach we adopted is to utilize the expression of r 1 ( θ ) .

From d r 1 / d θ = f 1 ( r 0 , θ ) , we obtain (16) r 1 ( θ ) = 0 θ f 1 ( r 0 , θ ) d θ + c 1 .

By noting that f 1 ( r 0 , θ ) is a periodic function, according to Lemma 2, we know (17) 0 θ f 1 ( r 0 , θ ) d θ = g 1 θ + φ 1 ( r 0 , θ ) , where g 1 = ( 1 / 2 π ) 0 2 π f 1 ( r 0 , θ ) d θ g ¯ 1 ( r 0 ) , and φ 1 ( r 0 , θ ) is periodic function with period 2 π .

From Lemma 1, we know that if r = r ( θ ) is a limit cycle of system (9), then r ( θ ) and r 1 ( θ ) in (16) are periodic functions, too.

So from Lemma 2, we know g 1 = g ¯ 1 ( r 0 ) = 0 . By solving that algebraic equation, we can determine the value of constant r 0 .

Remark 4.

In fact, the function g ¯ 1 ( r 0 ) is closely related to the first order Poincare-Melnikov integral of the perturbed system (9) near close orbit of unperturbed system (9) | ɛ = 0 .

In detail, g ¯ 1 ( r 0 ) = - ( 1 / 2 π r 0 ) L q 1 ( x , y ) d x - p 1 ( x , y ) d y , where close orbit L : x 2 + y 2 = r 0 2 . From [8, 9], we know - 2 π r 0 g ¯ 1 ( r 0 ) is the first order Poincare-Melnikov integral. So the zeros of g ¯ 1 ( r 0 ) are closely related to the number and position of limit cycles of the perturbed system (9).

It should be pointed out that the function g ¯ 1 ( r 0 ) is also closely related to the first order averaging of 1-dimensional 2 π -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 <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M148"> <mml:msub> <mml:mrow> <mml:mi>r</mml:mi></mml:mrow> <mml:mrow> <mml:mn>1</mml:mn></mml:mrow> </mml:msub> <mml:mo mathvariant="bold">(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo mathvariant="bold">)</mml:mo></mml:math> </inline-formula>

Substitute the value of r 0 into (16); we can obtain expression of φ 1 ( r 0 , θ ) . Thus we obtain (18) r 1 ( θ ) = φ 1 ( r 0 , θ ) + c 1 .

To determine the value of c 1 , new algorithm proposed in this paper needs the expression of r 2 ( θ ) . From d r 2 / d θ = f 2 ( r 1 , r 0 , θ ) , we get (19) r 2 ( θ ) = 0 θ f 2 ( r 1 , r 0 , θ ) d θ + c 2 .

From Lemma 2, we know (20) 0 θ f 2 ( r 1 , r 0 , θ ) d θ = g 2 θ + φ 2 ( r 1 , r 0 , θ ) , where g 2 = ( 1 / 2 π ) 0 2 π f 2 ( r 1 , r 0 , θ ) d θ g ¯ 2 ( c 1 ) .

From the fact that r 2 ( θ ) is a periodic function and Lemma 2, we get g ¯ 2 ( c 1 ) = 0 .

By solving the above algebraic equation, we determine the value of c 1 . Thus we have obtained r 1 ( θ ) by (18).

2.3. Determination of <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M162"> <mml:msub> <mml:mrow> <mml:mi>r</mml:mi></mml:mrow> <mml:mrow> <mml:mi>k</mml:mi></mml:mrow> </mml:msub> <mml:mo mathvariant="bold">(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo mathvariant="bold">)</mml:mo></mml:math> </inline-formula>

Assuming that we have obtained the explicit expressions of r 0 , r 1 ( θ ) , , r k - 1 ( θ ) , now we start to determine r k ( θ ) .

From d r k / d θ = f k ( r k - 1 , r k - 2 , , r 1 , r 0 , θ ) , we get (21) r k ( θ ) = 0 θ f k ( r k - 1 , r k - 2 , , r 1 , r 0 , θ ) d θ + c k .

To determine the expression of r k ( θ ) is to determine the value of c k . According to the algorithm proposed in this paper, we resort to the expression of r k + 1 ( θ ) . From d r k + 1 / d θ = f k + 1 ( r k , r k - 1 , , r 1 , r 0 , θ ) , we get (22) r k + 1 ( θ ) = 0 θ f k + 1 ( r k , r k - 1 , , r 1 , r 0 , θ ) d θ + c k + 1 .

From Lemma 2, we know (23) 0 θ f k + 1 ( r k , r k - 1 , , r 1 , r 0 , θ ) d θ = g k + 1 θ + φ k + 1 ( r k , r k - 1 , , r 0 , θ ) , where (24) g k + 1 = 1 2 π 0 2 π f k + 1 ( r k , r k - 1 , , r 0 , θ ) d θ g ¯ k + 1 ( c k ) .

Because r k + 1 ( θ ) is a periodic function, from Lemma 2, we get that g k + 1 = g ¯ k + 1 ( c k ) = 0 .

By solving the algebraic equation, we obtain the value of c k ; thus we determine r k ( θ ) by (20).

Thus we can compute the shape of limit cycles of system (9) to any given order of ɛ 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 (25) x = y , y = - x + ɛ ( 1 - x 2 ) y up to o ( ε 7 ) .

First we make a polar coordinates transformation x = r cos ( θ ) y = r sin ( θ ) to system (25) and eliminate t ; then we can obtain (26) d r d θ = ɛ r ( 1 - r 2 cos 2 θ ) sin 2 θ - 1 + ɛ ( 1 - r 2 cos 2 θ ) sin θ cos θ .

Assume r ( θ ) = k = 0 ε k r k ( θ ) is the polar coordinates form of the limit cycles of (25) and substitute it into (26). By comparing first eight coefficients of terms ε k , k = 1,2 , , 8 in both sides of the above equation, we get (27) d r 0 d θ = 0 , d r 1 d θ = - r 0 ( 1 - r 0 2 cos 2 θ ) sin 2 θ f 1 ( r 0 , θ ) , d r 2 d θ = ( ( - 1 + r 0 2    cos 2 θ ) r 1 + 2 r 0 2 r 1 cos 2 θ ) si n 2 θ + ( - 1 + r 0 2 cos 2 θ ) r 0    × si n 2 θ ( - cos θ sin θ + r 0 2 sin θ cos 3 θ ) f 2 ( r 1 , r 0 , θ ) , d r 8 d θ f 8 ( r 7 , r 6 , , r 0 , θ ) . Here for long expressions, the formula of f k ( r k - 1 , , r 1 , r 0 , θ ) ,   k = 3,4 , , 8 is omitted.

From d r 0 / d t = 0 , we get that r 0 is arbitrary constant.

To determine the value of r 0 , we compute the following expression of r 1 ( θ ) : (28) r 1 ( θ ) = 0 θ f 1 ( r 0 , θ ) d θ + c 1 = g 1 θ + φ 1 ( r 0 , θ ) + c 1 , where (29) g 1 = 1 2 π 0 2 π f 1 ( r 0 , θ ) d θ = - r 0 ( 1 - r 0 2 cos 2 θ ) sin 2 θ d θ = 1 8 r 0 3 - 1 2 r 0 . Because g 1 = 0 , so we get r 0 = 2 .

Substitute r 0 = 2 into (28); we obtain (30) r 1 ( θ ) = 0 θ f 1 ( 2 , θ ) d θ + c 1 = 2 cos θ sin θ - 2 cos 3 θ sin θ + c 1 .

To determine the value of c 1 , we compute the expression of r 2 ( θ ) : (31) r 2 ( θ ) = 0 θ f 2 ( r 1 , r 0 , θ ) d θ + c 2 = g 2 θ + φ 2 ( r 1 , r 0 ) + c 2 , where (32) g 2 = 1 2 π 0 2 π f 2 ( 2 cos θ sin θ - 2 cos 3 θ sin θ + c 1 , 2 , θ ) d θ = c 1 .

Because g 2 = 0 , so we get c 1 = 0 .

Thus explicit expression of r 1 ( θ ) is given by (30) with c 1 = 0 .

Substitute r 1 ( θ ) and r 0 = 2 into (31); we obtain (33) r 2 ( θ ) = 0 θ f 2 ( r 1 , r 0 , θ ) d θ + c 2 = 2 cos 2 θ - 23 2 cos 4 θ + 49 3 cos 6 θ - 7 cos 8 θ + 1 6 + c 2 . In a similar way, we determine the value of c 2 and obtain following results: (34) r 2 ( θ ) = 2 cos 2 θ - 23 2 cos 4 θ + 49 3 cos 6 θ - 7 cos 8 θ + 17 96 , r 3 ( θ ) = 1 2304 ( ( 9 θ ) 486 cos θ + 544 cos ( 3 θ ) - 24 cos ( 5 θ ) r 3 ( θ ) = g h h h h - 621 cos ( 7 θ ) - 297 cos ( 9 θ ) ) sin 3 θ , r 4 ( θ ) = - 4433 6 cos 14 θ + 715 4 cos 16 θ - 1577 552960 r 4 ( θ ) = - 85021 1728 cos 6 θ + 21505 18 cos 12 θ + 17 96 cos 2 θ r 4 ( θ ) = + 203747 576 cos 8 θ - 28153 30 cos 10 θ - 1045 1152 cos 4 θ , r 5 ( θ ) = - 4199 4096 cos 9 ( 2 θ ) sin ( 2 θ ) + 1105 12288 cos 8 ( 2 θ ) r 5 ( θ ) = × sin ( 2 θ ) - 16237 46080 cos 6 ( 2 θ ) sin ( 2 θ ) r 5 ( θ ) = + 412037 921600 sin ( 2 θ ) cos 4 ( 2 θ ) r 5 ( θ ) = - 105841 518400 cos 2 ( 2 θ ) sin ( 2 θ ) r 5 ( θ ) = + 623977 33177600 sin ( 2 θ ) + 169 48 cos 7 ( 2 θ ) r 5 ( θ ) = × sin ( 2 θ ) - 18203 49152 cos ( 2 θ ) sin ( 2 θ ) r 5 ( θ ) = + 30479 13824 sin ( 2 θ ) cos 3 ( 2 θ ) r 5 ( θ ) = - 797999 184320 cos 5 ( 2 θ ) sin ( 2 θ ) , r 6 ( θ ) = - 44679949 77414400 cos 7 ( 2 θ ) - 19738393 398131200 cos 3 ( 2 θ ) r 6 ( θ ) = + 16357283 55296000 cos 5 ( 2 θ ) - 2261 16384 cos 11 ( 2 θ ) r 6 ( θ ) = - 10462939 176947200 + 207757 442368 cos 9 ( 2 θ ) + 215852651 16588800 r 6 ( θ ) = × cos 6 ( 2 θ ) - 52003 32768 cos 12 ( 2 θ ) + 5393 26542080 r 6 ( θ ) = × cos ( 2 θ ) + 166171249 132710400 cos 2 ( 2 θ ) r 6 ( θ ) = - 8240753 589824 cos 8 ( 2 θ ) + 122417 16384 cos 10 ( 2 θ ) r 6 ( θ ) = - 1624125463 265420800 cos 4 ( 2 θ ) , r 7 ( θ ) = - 841356266407 2926264320000 cos 2 ( 2 θ ) sin ( 2 θ ) r 7 ( θ ) = + 298012397 11796480 cos 9 ( 2 θ ) sin ( 2 θ ) r 7 ( θ ) = + 3520666153 743178240    cos 8 ( 2 θ ) sin ( 2 θ ) r 7 ( θ ) = - 14166301337 3612672000 cos 6 ( 2 θ ) sin ( 2 θ ) r 7 ( θ ) = + 2104280130881 1300561920000 sin ( 2 θ ) cos 4 ( 2 θ ) r 7 ( θ ) = + 334305 131072 cos 13 ( 2 θ ) sin ( 2 θ ) + 260015 393216 r 7 ( θ ) = × cos 12 ( 2 θ ) sin ( 2 θ ) - 7451287 589824 cos 11 ( 2 θ ) sin ( 2 θ ) r 7 ( θ ) = - 6238099 2211840 cos 10 ( 2 θ ) sin ( 2 θ ) r 7 ( θ ) = + 283692942563 23410114560000 sin ( 2 θ ) - 533135791 20643840 r 7 ( θ ) = × cos 7 ( 2 θ ) sin ( 2 θ ) + 8763093 26214400 cos ( 2 θ ) sin ( 2 θ ) r 7 ( θ ) = - 5835758083 1592524800 sin ( 2 θ ) cos 3 ( 2 θ ) + 74189164357 5308416000 r 7 ( θ ) = × cos 5 ( 2 θ ) sin ( 2 θ ) .

So the asymptotic expansion of limit cycle of system (25) for ɛ > 0 and small is the following: (35) r ( θ ) = k = 0 7 ε k r k ( θ ) + o ( ε 7 ) .

The first seven terms in the above expansion of r ( θ ) are similar to the ones given in Section 3 of . Here we present the expression of r 7 ( θ ) 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 ɛ = 0 , ( 1 / 5 ) , ( 1 / 2 ) , ( 9 / 10 ) are plotted in Figure 1. The periodic orbit x 2 + y 2 = 4 of system (25) for ɛ = 0 is drawn in solid line, the limit cycle of system (25) for ɛ = 1 / 5 is drawn in dashed line, the limit cycle of system (25) for ɛ = 1 / 2 is drawn in solid line, and the limit cycle of system (25) for ɛ = 9 / 10 is drawn in dotted line.

The periodic orbit x 2 + y 2 = 4 and the analytical approximations of the limit cycle of Van der Pol system (25), for the values ɛ = 0 , ( 1 / 5 ) , ( 1 / 2 ) , ( 9 / 10 ) .

4. The Shape of Limit Cycle of Perturbations of a System Having Nonlinear Isochronous Center

Consider the following perturbed system: (36) x = - y + 2 x y - 2 y 3 + ɛ ( x 3 - 3 x ) , y = x - y 2 . From , we know that as ɛ = 0 , nonlinear system (36) has isochronous center O ( 0,0 ) . 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: (37) x = u + v 2 , y = v , t = - τ to system (36) and get (38) d u d τ = v + ɛ ( 3 u + 3 v 2 - ( u + v 2 ) 3 ) , d v d τ = - u .

4.1. The Shape of Limit Cycle of the Perturbed System (<xref ref-type="disp-formula" rid="EEq26">38</xref>)

In this subsection we start to compute the analytic expansion of the limit cycle of the perturbed system (38) to the second order of ɛ .

First let u = r cos θ , v = r sin θ ; then system (38) is transformed into the following polar coordinate form: (39) d r d θ = ɛ ( 3 r cos θ + 3 r 2 sin 2 θ - ( r cos θ + r 2 sin 2 θ ) 3 ) cos θ × ( - 1 + ɛ ( ( r cos θ + r 2 sin 2 θ ) 2 ) sin θ - 3 cos θ - 3 r sin 2 θ + ( cos θ + r sin 2 θ ) ghhhhhhhhh × ( r cos θ + r 2 sin 2 θ ) 2 ) sin θ ) - 1 . According to similar process in Section 3, we get (40) g 1 = 3 16 r 0 ( r 0 4 + 2 r 0 2 - 8 ) .

By solving g 1 = 0 , we get the positive solution r 0 = 2 .

From ( d ( - 2 π r 0 g 1 ) / d r 0 ) | r 0 = 2 = - 9 2 π < 0 and , we conclude that as ɛ > 0 , system (38) has a stable limit cycle, denoted by Γ 1 , near the close curve u 2 + v 2 = 2 on the phase plane.

By applying the algorithm described in Section 2 to system (39), we get the asymptotic expansion of stable limit cycle Γ 1 of system (38), (41) r ( θ ) = r 0 + ɛ r 1 ( θ ) + ε 2 r 2 ( θ ) + O ( ε 3 ) , where r 0 , r 1 ( θ ) , r 2 ( θ ) are given in the following: (42) r 0 = 2 , r 1 ( θ ) = 1 35 sin ( 26 + 70 2 cos 5 θ - 105 2 cos 3 θ g h g h h h - 22 cos 2 θ - 40 cos 6 θ + 36 cos 4 θ ) , r 2 ( θ ) = - 914 35 cos 3 θ + 14858 175 cos 5 θ + 17408 105 cos 9 θ - 7188 49 cos 7 θ + 192 7 cos 13 θ - 1168 11 cos 11 θ - 208 49 2 cos 14 θ - 258 35 2 cos 12 θ + 12312 175 2 × cos 10 θ - 8367 70 2 cos 8 θ + 1037 14 2 cos 6 θ - 927 140 2 cos 4 θ - 269 35 2 cos 2 θ + 1130351 940800 2 .

In Figure 2 we illustrate the shape of the limit cycle Γ 1 of the system (38) by using formula (41) for the values ɛ = 1 / 20 . The periodic orbit u 2 + v 2 = 2 of system (38) for ɛ = 0 is drawn in solid line, and the limit cycle Γ 1 is drawn in dash line. We have also plotted the limit cycle Γ 1 for the value ɛ = 1 / 20 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.

The periodic orbit u 2 + v 2 = 2 for ɛ = 0 and the analytical and numerical approximations of the limit cycle for ɛ = 1 / 20 of system (38).

4.2. The Shape of Limit Cycle of the Original Perturbed System (<xref ref-type="disp-formula" rid="EEq24">36</xref>)

In this subsection, we give the analytic expansion of the limit cycle of perturbed system (36) to the second order of ɛ .

Rewrite the limit cycle Γ 1 of system (38) into the following parametric form: (43) u = r ( θ ) cos θ , v = r ( θ ) sin θ , where r ( θ ) 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 ɛ = ( 1 / 20 ) is unstable and its parametric form is the following: (44) x = u + v 2 = r ( θ ) cos θ + ( r ( θ ) sin θ ) 2 , y = v = r ( θ ) sin θ . The shape of limit cycle of the system (25) for ɛ = 1 / 20 is plotted by using formula (44) in Figure 3.

The periodic orbit for ɛ = 0 and the analytical and numerical approximations of the limit cycle for ɛ = 1 / 20 of system (36).

In Figure 3, the periodic orbit ( x - y 2 ) 2 + y 2 = 2 of unperturbed system (36) for ɛ = 0 is drawn in sold line, and the limit cycle of the perturbed system (36) for ɛ = 1 / 20 is drawn in dash line.

Conflict of Interests

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

Acknowledgments

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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