A quintic Z2-equivariant Li\'enard system arising from the complex Ginzburg-Landau equation: (II)

TL;DR

This paper investigates the global dynamics of a specific quintic Z2-equivariant Li'enard system derived from the complex Ginzburg-Landau equation, focusing on the case where the sum of equilibrium indices is positive.

Contribution

It extends previous analysis by studying the system's behavior when the sum of equilibrium indices is positive, revealing new global dynamics.

Findings

Characterization of equilibrium configurations for a positive index sum

Identification of bifurcation scenarios in the system

Description of long-term behaviors of solutions

Abstract

We continue to study a quintic Z2-equivariant Li\'enard system $\dot x=y,\dot y=-(a_0x+a_1x^3+a_2x^5)-(b_0+b_1x^2)y$ with $a_2b_1\ne 0$, 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 Li\'enard system when the sum of the indices of all equilibria is $1$, i.e., $a_2>0$.