dorsal/arxiv
View SchemaTwisted and Nontwisted Bifurcations Induced by Diffusion
| Authors | Xiao-Biao Lin |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9604004 |
| URL | https://arxiv.org/abs/patt-sol/9604004 |
| DOI | 10.1007/BF02218759 |
Abstract
We discuss a diffusively perturbed predator-prey system. Freedman and Wolkowicz showed that the corresponding ODE can have a periodic solution that bifurcates from a homoclinic loop. When the diffusion coefficients are large, this solution represents a stable, spatially homogeneous time-periodic solution of the PDE. We show that when the diffusion coefficients become small, the spatially homogeneous periodic solution becomes unstable and bifurcates into spatially nonhomogeneous periodic solutions. The nature of the bifurcation is determined by the twistedness of an equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients decrease. In the nontwisted case two spatially nonhomogeneous simple periodic solutions of equal period are generated, while in the twisted case a unique spatially nonhomogeneous double periodic solution is generated through period-doubling. Key Words: Reaction-diffusion equations; predator-prey systems; homoclinic bifurcations; periodic solutions.
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"abstract": "We discuss a diffusively perturbed predator-prey system. Freedman and\nWolkowicz showed that the corresponding ODE can have a periodic solution that\nbifurcates from a homoclinic loop. When the diffusion coefficients are large,\nthis solution represents a stable, spatially homogeneous time-periodic solution\nof the PDE. We show that when the diffusion coefficients become small, the\nspatially homogeneous periodic solution becomes unstable and bifurcates into\nspatially nonhomogeneous periodic solutions.\n The nature of the bifurcation is determined by the twistedness of an\nequilibrium/homoclinic bifurcation that occurs as the diffusion coefficients\ndecrease. In the nontwisted case two spatially nonhomogeneous simple periodic\nsolutions of equal period are generated, while in the twisted case a unique\nspatially nonhomogeneous double periodic solution is generated through\nperiod-doubling.\n Key Words: Reaction-diffusion equations; predator-prey systems; homoclinic\nbifurcations; periodic solutions.",
"arxiv_id": "patt-sol/9604004",
"authors": [
"Xiao-Biao Lin"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1007/BF02218759",
"title": "Twisted and Nontwisted Bifurcations Induced by Diffusion",
"url": "https://arxiv.org/abs/patt-sol/9604004"
},
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