Equivariant Hopf bifurcation with arbitrary pressure laws

TL;DR

This paper investigates the dynamics of equivariant Hopf bifurcation in PDE systems, linking it to hyperbolic conservation laws and generalizing previous $O(2)$ bifurcation studies with two parameters.

Contribution

It introduces a generalized framework for $O(2)$ Hopf bifurcation in PDEs, utilizing center manifold reduction to analyze parameter-dependent bifurcation behavior.

Findings

Establishes connections between bifurcation dynamics and hyperbolic conservation laws

Provides a class of $O(2)$ Hopf bifurcations with two parameters in PDE systems

Uses center manifold approximation to determine key parameters in normal form

Abstract

The equivariant Hopf bifurcation dynamics of a class of system of partial differential equations is carefully studied. The connections between the current dynamics and fundamental concepts in hyperbolic conservation laws are explained. The unique approximation property of center manifold reduction function is used in the current work to determine certain parameter in the normal form. The current work generalizes the study of the second author ([J. Yao, $O(2)$-Hopf bifurcation for a model of cellular shock instability, Physica D, 269 (2014), 63-75.]) and supplies a class of examples of $O(2)$ Hopf bifurcation with two parameters arising from systems of partial differential equations.