The Critical Phase Curve of Van Der Pol Equation

JournaJ ofTaiwan NormaJ University: Mathematics, Science & TechnoJogy 2001,46(1,2) , 1-12

The Critical Phase Curve of Van Der Pol Equation

Je-Chiang Tsai Tai-Yih Tso

Department of Mathematics‘National Taiwan Normal University

Abstract

This article is concerned w出 thespecial trajectory y咱 (ιμ) which is the leadi 月 termof the asymptotic

叫山on of Van der Pol eq叫IOn X叫抖'(x2 -1) + x = Oin the phase plane for some region. We show that in the phase plane, the difference of this asymptotic solution and the Iimit cycle of Van der Pol equation is not greater than O(

,L1-

1 ! 3 )的 μ →+∞ for aIl -1 三 X 三 O. Using 帥的ult， we can show 伽t every

tr句 ectoryof Van der Pol equation starting from y-axis with initial value bigger than that of the limit cycle gets close tω0 伽Ii m圳I甘1

μ 一吾+∞

Key words: Van der Pol equation, limit cyclc

1 Introduction

Let us consider the well-know Van der Pol

equation x"+μx'(x2 -1)+x=O ， where x'=dx/dt. It is easily to see that the Van der Pol equation is equivalent to the fol1owing equation

、‘， j 4 , sv fa 胃、

y

、‘'，/

-A

+ 勻，但 、、.，'， 4 ,', ',‘\ x r'. ‘、 μ + 、、_.. ',', 、 JUυ hox

y--一一一 、‘.，/、、 •. ', 4 ，ι 侈， ι '，.‘、、， s ﹒‘、、

xve

rl--ldIl--K

in the phase plane, called the Van der Pol equation in the phase plane.

The study of the unique periodic solution of the Van der Pol equation was started by Van der Pol. The behavior of the periodic solution of the Van der Pol equation exhibits a relaxation oscil1ation as the parameterμ>>1 . Therefore, there are many researchers investigating the asymptotic behavior and the numerical solution of the periodic solution of the Van der Pol equation withμ>>1 (Ponzo and wax,1965; Andersen and Geer, 1982).

In this article, the asymptotic method was used to analysis the phase path of the Van der Pol equation in the phase plane. We define a special trajectory, called the critical phase curve y ∞ (x; μ) which is the solution of the following scalar equation

企=一主 +μ(1-

x2)

ax

y

The critical phase curve y∞ (x; μ) is related to the asymptotic behavior of yp(X; μwhich is the corresponding phase path ofthe unique periodic solution of the Van der Pol equation. There are three major work in this study as fol1owing:

(1) y∞(一1;μ) - yp(一 1) = O(μ 一 113 ) . This result

confirms that the leading term of the asymptotic solution y∞ (X; μis not only a good asymptotic solution up to x =' - } +μI/3 ， but also to x= 一l.

(2) Y∞ (x; μ) 一 Yp(x; μ)=' O(μ-113) for al1一 l 三 X 三 oas μ 一~+∞.

(3) In the phase plane, every phase path of Van der Pol equation, say y(X; μ) ， with y(O; μ) > yp (0; μ) , gets close to the limit cycle from its first time on intersecting x=l in the fourth quadrant.

This article is organized in the fol1owing structure. In section 2, we prove the existence and the uniqueness of the critical phase curve y∞ (x; μ) for all μ> O. And then, we estimate y島 (x; μ) ， for -1 至 X 三三

0

, and μ>>1 ， in section 3. Moreover

,

we also prove that

Y∞ (x; μ)- yp(x; μ) =' O(μ113

),

for all 一 1 三 x 豆 O asμ →+∞. Finally, we have a concluding remark in section 4.