Bifurcations and Symmetry in Dissipative Dynamics

This work consists of applications of bifurcation theory to nonlinear dynamical systems. We begin by analyzing three instabilities that may occur in dynamical systems governed by dissipative nonlinear differential equations. One instability occurs when a fixed point with a breve {rm S}il'nikov homoclinic orbit undergoes a Hopf bifurcation. In an application this instability is found to give rise to unusual chaotically reversing waves. Next we turn to instabilities which possess spatial symmetry and analyze mode interactions involving localized solutions to a reaction-diffusion system. We first consider a single O(2) symmetric disk-shaped solution which loses stability simultaneously to radial oscillations and steady deformations with azimuthal wave number n = 2. Then we consider a stripe solution with O(2) times Z_2 symmetry which, as a system parameter changes, loses stability to zigzag and varicose perturbations. The analysis is based solely on symmetry considerations and the results therefore have potential applications in many other systems. Finally, we examine the effects of small imperfections in the symmetry of a system on resulting bifurcations. We investigate the effects of distant endwalls on both the steady-state and oscillatory instabilities of a translation invariant state. In particular we numerically investigate the O(2) equivariant Hopf normal form with terms which break rotation symmetry. The resulting homoclinic chaos resembles closely the dynamics observed in binary fluid convection experiments.