dorsal/arxiv
View SchemaMulti-Phase Patterns in Periodically Forced Oscillatory Systems
| Authors | Christian Elphick, Aric Hagberg, Ehud Meron |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9902006 |
| URL | https://arxiv.org/abs/patt-sol/9902006 |
| DOI | 10.1103/PhysRevE.59.5285 |
Abstract
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 $\pi/2$. 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: $\pi$-fronts separating states with a phase shift of $\pi$ and $\pi/2$-fronts separating states with a phase shift of $\pi/2$. We find a new type of front instability where a stationary $\pi$-front ``decomposes'' into a pair of traveling $\pi/2$-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 $\pi/2$-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 $\pi$-fronts decompose into n traveling $\pi/n$-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.
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"abstract": "Periodic forcing of an oscillatory system produces frequency locking bands\nwithin which the system frequency is rationally related to the forcing\nfrequency. We study extended oscillatory systems that respond to uniform\nperiodic forcing at one quarter of the forcing frequency (the 4:1 resonance).\nThese systems possess four coexisting stable states, corresponding to uniform\noscillations with successive phase shifts of $\\pi/2$. Using an amplitude\nequation approach near a Hopf bifurcation to uniform oscillations, we study\nfront solutions connecting different phase states. These solutions divide into\ntwo groups: $\\pi$-fronts separating states with a phase shift of $\\pi$ and\n$\\pi/2$-fronts separating states with a phase shift of $\\pi/2$. We find a new\ntype of front instability where a stationary $\\pi$-front ``decomposes\u0027\u0027 into a\npair of traveling $\\pi/2$-fronts as the forcing strength is decreased. The\ninstability is degenerate for an amplitude equation with cubic nonlinearities.\nAt the instability point a continuous family of pair solutions exists,\nconsisting of $\\pi/2$-fronts separated by distances ranging from zero to\ninfinity. Quintic nonlinearities lift the degeneracy at the instability point\nbut do not change the basic nature of the instability. We conjecture the\nexistence of similar instabilities in higher 2n:1 resonances (n=3,4,..) where\nstationary $\\pi$-fronts decompose into n traveling $\\pi/n$-fronts. The\ninstabilities designate transitions from stationary two-phase patterns to\ntraveling 2n-phase patterns. As an example, we demonstrate with a numerical\nsolution the collapse of a four-phase spiral wave into a stationary two-phase\npattern as the forcing strength within the 4:1 resonance is increased.",
"arxiv_id": "patt-sol/9902006",
"authors": [
"Christian Elphick",
"Aric Hagberg",
"Ehud Meron"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevE.59.5285",
"title": "Multi-Phase Patterns in Periodically Forced Oscillatory Systems",
"url": "https://arxiv.org/abs/patt-sol/9902006"
},
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