A Quintic \(\boldsymbol{\mathbb{Z}_2}\)-Equivariant Liénard System Arising from the Complex Ginzburg–Landau Equation

In this paper, we study a quintic Liénard system with -equivariance, arising from the complex Ginzburg–Landau equation. Although this system is a versal unfolding of the germ near the origin, it cannot be changed equivalently into a near-Hamiltonian system for global variables and parameters so that its dynamics cannot be studied via counting the isolate zeros of Abelian integrals as usual. We present a complete study of this system with , i.e., the sum of indices of equilibria is , and show that this system exhibits at most three limit cycles and a double center. The necessary and sufficient conditions are obtained on the existence of three limit cycles, a stable two-saddle heteroclinic loop, an unstable figure-eight loop, and two stable homoclinic loops. A global bifurcation diagram and the corresponding global phase portraits in the Poincaré disc of this system are given, including pitchfork bifurcation, Hopf bifurcation, transcritical bifurcation, two-saddle heteroclinic loop bifurcation, double limit cycle bifurcation, homoclinic bifurcation, saddle connection bifurcation, and degenerate Bogdanov–Takens bifurcation. Note that the dynamics of this quintic Liénard system is so complicated that it has infinitely many bifurcation surfaces of saddle connection.