Multiphase patterns in periodically forced oscillatory systems.

Periodic forcing of an oscillatory system produces frequency locking bands within which the system frequency is rationally related to the forcing frequency. We study extended oscillatory systems that respond to uniform periodic forcing at one quarter of the forcing frequency (the 4:1 resonance). These systems possess four coexisting stable states, corresponding to uniform oscillations with successive phase shifts of pi/2. Using an amplitude equation approach near a Hopf bifurcation to uniform oscillations, we study front solutions connecting different phase states. These solutions divide into two groups: pi fronts separating states with a phase shift of pi and pi/2 fronts separating states with a phase shift of pi/2. We find a type of front instability where a stationary pi front "decomposes" into a pair of traveling pi/2 fronts as the forcing strength is decreased. The instability is degenerate for an amplitude equation with cubic nonlinearities. At the instability point a continuous family of pair solutions exists, consisting of pi/2 fronts separated by distances ranging from zero to infinity. Quintic nonlinearities lift the degeneracy at the instability point but do not change the basic nature of the instability. We conjecture the existence of similar instabilities in higher 2n:1 resonances (n=3,4, em leader) where stationary pi fronts decompose into n traveling pi/n fronts. The instabilities designate transitions from stationary two-phase patterns to traveling 2n-phase patterns. As an example, we demonstrate with a numerical solution the collapse of a four-phase spiral wave into a stationary two-phase pattern as the forcing strength within the 4:1 resonance is increased.