Global Dynamics and Periodic-Annulus Bifurcations in a Reversible Quintic Liénard System

Abstract

We study the global dynamics and bifurcations of the reversible quintic Liénard system $\dot{x}=y$, $\dot{y}=a_0x+a_1x^3+a_2x^5+y(b_0x+b_1x^3)$, with $a_2\ne0$. In the original five-parameter space we treat both signs of $a_2$ and determine all global phase-portrait types on the Poincaré disc with one, three, or five finite equilibria. The analysis covers generic and generalized pitchfork, saddle-node, Bogdanov--Takens, and Hopf bifurcations; nilpotent and Hamiltonian degeneracies; bifurcations at infinity; and homoclinic, heteroclinic, limit-cycle-breaking, and periodic-annulus-breaking transitions. For a parameterized class of generalized polynomial Liénard systems whose unique finite equilibrium is the nilpotent origin with exactly one elliptic and one hyperbolic sector we establish a unique critical value of the higher-order damping parameter in the intermediate-degree range, nonexistence of a large-scale periodic annulus in the critical-degree case, and its persistence when the restoring degree dominates. Here large-scale means that the annulus closure meets the boundary of the Poincaré disc. In the degree range relevant to the quintic system the critical value gives a large-scale periodic-annulus--homoclinic-orbit-family bifurcation marked by a unique separatrix connection to infinity. For the quintic system with multiple finite equilibria periodic-annulus-breaking bifurcations arise from finite separatrix connections or separatrices reaching equilibria at infinity. Using Poincaré compactification, local normal forms, rotated vector fields, and separatrix arguments we describe bifurcation sets algebraically or by uniquely characterized critical functions. We obtain complete fixed-coefficient parameter partitions outside $Ω_{14}$ and determine the admissible critical-surface configurations and their phase portraits inside it.