Applied Mathematics and Plasma Physics

Christian Elphick, Aric Hagberg and Ehud Meron, "Multiphase patterns in periodically forced oscillatory systems", *Phys. Rev. E*, vol. 59, no. 5, pp. 5285--5291, 1999

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
*π/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: *π*-fronts separating
states with a phase shift of *π* and
*π/2*-fronts separating states with a phase
shift of *π/2*. We find a new type of front
instability where a stationary *π*-front
``decomposes'' into a pair of traveling
*π/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
*π/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,..*) where stationary
*π*-fronts decompose into *n*
traveling *π/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.

@article{elphick-1999-multiphase,

author = {Christian Elphick and Aric Hagberg and Ehud Meron},

title = {Multiphase patterns in periodically forced oscillatory systems},

year = {1999},

urlpdf = {http://math.lanl.gov/~hagberg/Papers/fcgl2/multiphase.pdf},

journal = {Phys. Rev. E},

volume = {59},

number = {5},

pages = {5285--5291}

}