  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.
