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June 2019 Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System
Shunsuke KOBAYASHI, Takashi Okuda SAKAMOTO
Tokyo J. Math. 42(1): 121-183 (June 2019). DOI: 10.3836/tjm/1502179295

Abstract

We study the bifurcations of small amplitude time-periodic solutions and chaotic solutions of a two-component integro-differential reaction-diffusion system in one spatial dimension. The system has doubly degenerate points and triply degenerate points. The following results are obtained. (I) Around the doubly degenerate points, a reduced two-dimensional dynamical system on the center manifold is obtained. We find that the small amplitude stable time-periodic solutions can bifurcate from the non-uniform stationary solutions through the Hopf bifurcations for all $n$. (II) Around the triply degenerate point, a three-dimensional dynamical system on the center manifold is obtained. The reduced system can be transformed into normal form for the Hopf-Pitchfork bifurcation. The truncated normal form can possess the invariant tori and the heteroclinic loop. Furthermore, the system under the non $S^{1}$-symmetric perturbation may possess the Shil'nikov type homoclinic orbit. Numerical results for the integro-differential reaction-diffusion system are presented and found to be convincing.

Citation

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Shunsuke KOBAYASHI. Takashi Okuda SAKAMOTO. "Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System." Tokyo J. Math. 42 (1) 121 - 183, June 2019. https://doi.org/10.3836/tjm/1502179295

Information

Published: June 2019
First available in Project Euclid: 18 July 2019

zbMATH: 07114904
MathSciNet: MR3982053
Digital Object Identifier: 10.3836/tjm/1502179295

Subjects:
Primary: 37G15
Secondary: 35B10, 35B32, 37D45, 37G05

Rights: Copyright © 2019 Publication Committee for the Tokyo Journal of Mathematics

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Vol.42 • No. 1 • June 2019
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