The Critical Phase Curve of Van Der Pol Equation

Abstract

本文探討Van der Pol 方程式在相平面上一條特殊臨界曲線，記為。它是Van der Pol 方程式在相平面上特定區域中對於極限環的漸進解。本研究證明在相平面的上半平面中，Van der Pol 方程式的極限環與臨界曲線之差至多為，當,。更進一步，可以利用這個結果，證明當時，相平面上任一條Van der Pol 方程式的解軌線從y軸出發且在極限環外部時，當第一次與x=1相交於第四象限之後，其與極限環的差至多為。

This article is concerned with the special trajectorywhich is the leading term of the asymptotic solution of Van der Pol equationin the phase plane for some region. We show that in the phase plane, the difference of this asymptotic solution and the limit cycle of Van der Pol equation is not greater thanasfor all. Using this result, we can show that every trajectory of Van der Pol equation starting from y-axis with initial value bigger than that of the limit cycle gets close to the limit cycle byfrom its first time on intersecting x=1 in the four quadrant as.