Bifurcations of Relative Equilibria

This paper discusses the dynamics and bifurcation theory of equivariant dynamical systemsnear relative equilibria, that is, group orbits invariant under the flow of an equivariant vector field. The theory developed here applies, in particular, to secondary steady-state bifurcations from invariant equilibria. Let $\Gamma $ be a compact group of symmetries of $R^n $ and let $x_0 $ be in $R^n $. Suppose that f is a smooth $\Gamma $-equivariant vector field and $\Sigma $ the isotropy group of $x_0 $. It is shown that there exists a $\Sigma $-equivariant vector field $f_N $, defined on the space normal to X at $x_0 $, and that the local asymptotic dynamics of f are closely related to the local asymptotic dynamics of $f_N $. Next those bifurcations of X are studied which occur when an eigenvalue of $(df_N )_x $ crosses the imaginary axis. Properties of the vector field $f_N $ imply that branches of equilibria and periodic orbits of $f_N $ correspond to trajectories of f which are dense in tori. Field [Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), pp. 185–205] found bounds on the dimensions of these tori. Some of his results are extended. This theory is applied to the following specific problems: (1) Bifurcations of systems with $O(2)$ symmetry. (2) Bifurcations of steady-state solutions of the Kuramoto–Sivashinsky equation. (3) Secondary bifurcations in the planar Bénard problem.