Imperfection sensitivity of interacting hopf and steady-state bifurcations and their classification

Abstract

The bifurcation problem of interacting time-periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory. A complete classification, up to symmetry-covariant contact equivalence and codimension three, of generic perturbations of interacting Hopf and steady-state bifurcations is presented. The sensitivity of the bifurcation diagrams to imperfections is analyzed. Normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined. A variety of phenomena, such as gaps in Hopf branches, periodic motions not stably connected to steady states and the formation of islands, is discovered, which one can expect to find in perturbed evolution equations on pure geometric grounds. Implications for physical systems are discussed.