A quintic Z2-equivariant Liénard system arising from the complex Ginzburg-Landau equation: (II)

Abstract

We continue to study a quintic Z2-equivariant Liénard system $\dot{x}=y,\dot{y}=-(a_{0}x+a_1{x}^3+a_2{x}^5)-(b_0+b_1{x}^2)y$ with $a_2{b}_1\ne 0$, arising from the complex Ginzburg-Landau equation. Global dynamics of the system have been studied in [{\it SIAM J. Math. Anal.}, {\bf 55}(2023) 5993-6038] when the sum of the indices of all equilibria is $-1$, i.e., $a_2<0$. The aim of this paper is to study the global dynamics of this quintic Liénard system when the sum of the indices of all equilibria is $1$, i.e., $a_2>0$.

51 pages