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 [Formula Presented]. 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: [Formula Presented] fronts separating states with a phase shift of [Formula Presented] and [Formula Presented] fronts separating states with a phase shift of [Formula Presented]. We find a type of front instability where a stationary [Formula Presented] front “decomposes” into a pair of traveling [Formula Presented] 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 [Formula Presented] 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 [Formula Presented] resonances [Formula Presented] where stationary [Formula Presented] fronts decompose into n traveling [Formula Presented] fronts. The instabilities designate transitions from stationary two-phase patterns to traveling [Formula Presented]-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.