__Resonant
Standing-Wave Patterns in Forced Oscillations:__

__Mechanisms
and Forms.__

** Arik
Yochelis,** Aric Hagberg , Christian Elphick, Ehud Meron, Anna L. Lin,
and Harry L. Swinney

The subject of this study is an extended oscillatory system that
goes through a Hopf bifurcation to uniform oscillations and subjected to
uniform time periodic forcing at a frequency about twice as large as the Hopf frequency.
We use an amplitude equation, the forced CGL equation, to study resonant
patterns that oscillate at exactly half the forcing frequency. We find that
these patterns occupy a tongue-like domain in the plane spanned by the forcing
amplitude and frequency, and that this domain differs from the resonance tongue
domain of uniform oscillations. In particular, we find that resonant
standing-wave patterns persist outside the boundaries of the resonance tongue
of uniform oscillations. We explain this behavior by deriving and analyzing
amplitude equations for a Hopf-Turing bifurcation that exists in the forced CGL
equation. We also study the different mechanisms by which resonant standing
waves form inside and outside the resonant tongue of uniform oscillations.