Tokyo Journal of Mathematics

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

Shunsuke KOBAYASHI and Takashi Okuda SAKAMOTO

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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

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

First available in Project Euclid: 18 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37G15: Bifurcations of limit cycles and periodic orbits
Secondary: 37D45: Strange attractors, chaotic dynamics 35B10: Periodic solutions 35B32: Bifurcation [See also 37Gxx, 37K50] 37G05: Normal forms


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.

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  • \BibAuthorsDumortier F, Ibá$\tilde{\text{n}}$ez S, Kokubu H and Simó C, About the unfolding of a Hopf-zero singularity, Discrete and Continuous Dynamical Systems series A 33 (2013), 4435–4471.
  • \BibAuthorsFukusima S, Nakanishi S, Nakato Y and Ogawa T, Selection principle for various modes of spatially nonuniform electorochemical oscillations, The Journal of Chemical Physics 128, 014714 (2008), 1–10.
  • \BibAuthorsGuckenheimer J and Holmes P, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
  • \BibAuthorsHaragus M and Iooss G, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimentional Dynamics Systems, Springer, 2010.
  • \BibAuthorsKuznetsov Yu. A, Elements of Applied Bifurcation Theory, 3$^{{\text{rd}}}$ edition, Springer-Verlag, New York, 2004.
  • \BibAuthorsMoehlis J, Smith R. T, Holmes P and Faisst H, Models for turbulent plane Cuette flow using the proper orthogonal decomposition, Physics of Fluids 14 (7) (2002), 2493–2507.
  • \BibAuthorsOgawa T, Degenerate Hopf instability in oscillatory reaction-diffusion equations, DCDS Supplements, Special vol. (2007), 784–793.
  • \BibAuthorsOgawa T and Okuda T, Oscillatory dynamics in a reaction-diffusion system in the presence of 0:1:2 resonance, Networks and Heterogeneous Media 7 (4) (2012), 893–926.
  • \BibAuthorsOgawa T and Sakamoto O. T, Chaotic dynamics in an integro-differential reaction-diffusion system in the presence of 0:1:2 resonance, Mathematical Fluid Dynamics, Present and Future (2016), 531–562.
  • \BibAuthorsSmith R. T, Moehlis J, and Holmes P, Heteroclinic cycles and periodic orbits for the O(2)-equivariant 0:1:2 mode interaction, PHYSICA D 211 (2005), 347–376.
  • \BibAuthorsHolmes P, Unfolding a degenerate nonlinear oscillator: a codimension two bifurcation, Annals New York Academy of Sciences (1980), 473–488.