## Tokyo Journal of Mathematics

### Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System

#### 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.

#### Article information

Source
Tokyo J. Math., Volume 42, Number 1 (2019), 121-183.

Dates
First available in Project Euclid: 18 July 2019

https://projecteuclid.org/euclid.tjm/1563436917

Mathematical Reviews number (MathSciNet)
MR3982053

Zentralblatt MATH identifier
07114904

#### Citation

KOBAYASHI, Shunsuke; SAKAMOTO, Takashi Okuda. Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System. Tokyo J. Math. 42 (2019), no. 1, 121--183. https://projecteuclid.org/euclid.tjm/1563436917

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