Global dynamics of Liénard systems with arbitrary degrees

Abstract

The aim of this paper is to study global dynamics of Liénard systems with arbitrary degrees $\dot{x}=y$, $\dot{y}=a_{1}x+a_2{x}^{2m+1}+(a_3+a_4{x}^{2n-1})y$. The complex and rich dynamics are presented, in particular, including double limit cycle bifurcation, Hopf bifurcation, homoclinic bifurcation and heteroclinic bifurcation. We illustrate theoretical results by numerical simulations.