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 freq uency. We study extended oscillatory systems that respond to uniform period ic forcing at one quarter of the forcing frequency (the 4:1 resonance). The se systems possess four coexisting stable states, corresponding to uniform oscillations with successive phase shifts of pi/2. Using an amplitude equat ion approach near a Hopf bifurcation to uniform oscillations, we study fron t solutions connecting different phase states. These solutions divide into two groups: pi fronts separating states with a phase shift of pi and pi/2 f ronts separating states with a phase shift of pi/2. We find a type of front instability when a stationary pi front "decomposes" into a pair of traveli ng pi/2 fronts as the forcing strength is decreased. The instability is deg enerate for an amplitude equation with cubic nonlinearities. At the instabi lity point a continuous family of pair solutions exists, consisting of pi/2 fronts separated by distances ranging from zero to infinity. Quintic nonli nearities lift the degeneracy at the instability point but do not change th e basic nature of the instability. We conjecture the existence of similar i nstabilities in higher 2n:1 resonances (n = 3,4,...) where stationary pi fr onts decompose into n traveling pi/n fronts. The instabilities designate tr ansitions 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. [S1063-651X(99)06705-7].