On the determination of exact number of limit cycles in Lienard Systems

TL;DR

This paper provides a simplified proof for the exact number of limit cycles in Lienard systems under weaker conditions and offers improved amplitude estimates for the Van Der Pol oscillator, showing amplitude independence from asymptotic behavior.

Contribution

It introduces a simpler proof for limit cycle existence in Lienard systems with weaker assumptions and refines amplitude estimates for the Van Der Pol equation.

Findings

Simpler proof of limit cycle existence under weaker conditions

Improved amplitude estimates for Van Der Pol oscillator

Amplitude independence from asymptotic behavior of F

Abstract

We present a simpler proof of the existence of an exact number of one or more limit cycles to the Lienard system $\dot{x}=y-F(x) $, $\dot {y}=-g(xt)$, under weaker conditions on the odd functions $F(x) $ and $g(x) $ as compared to those available in literature. We also give improved estimates of amplitudes of the limit cycle of the Van Der Pol equation for various values of the nonlinearity parameter. Moreover, the amplitude is shown to be independent of the asymptotic nature of $F$ as $|x| \to\infty$.