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.