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Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 893-926. doi: 10.3934/nhm.2012.7.893
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Oscillatory dynamics in a reaction-diffusion system in the presence of 0:1:2 resonance

  • 1.
    Graduate school of Advanced Mathematical Science, Meiji University, Higashimita, 214-8571
  • 2.
    Meteorological college, Kashiwa, 277-0852
  • Received: 01 January 2012 Revised: 01 July 2012

  • Primary: 35B10, 37G05; Secondary: 37D45.

  • Oscillatory dynamics in a reaction-diffusion system with spatially nonlocal effect under Neumann boundary conditions is studied. The system provides triply degenerate points for two spatially non-uniform modes and uniform one (zero mode). We focus our attention on the 0:1:2-mode interaction in the reaction-diffusion system. Using a normal form on the center manifold, we seek the equilibria and study the stability of them. Moreover, Hopf bifurcation phenomena is studied for each equilibrium which has a Hopf instability point. The numerical results to the chaotic dynamics are also shown.

    Citation: Toshiyuki Ogawa, Takashi Okuda. Oscillatory dynamics in a reaction-diffusion system in the presence of 0:1:2 resonance[J]. Networks and Heterogeneous Media, 2012, 7(4): 893-926. doi: 10.3934/nhm.2012.7.893

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  • Oscillatory dynamics in a reaction-diffusion system with spatially nonlocal effect under Neumann boundary conditions is studied. The system provides triply degenerate points for two spatially non-uniform modes and uniform one (zero mode). We focus our attention on the 0:1:2-mode interaction in the reaction-diffusion system. Using a normal form on the center manifold, we seek the equilibria and study the stability of them. Moreover, Hopf bifurcation phenomena is studied for each equilibrium which has a Hopf instability point. The numerical results to the chaotic dynamics are also shown.


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  • This article has been cited by:

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    2. Shunsuke KOBAYASHI, Takashi Okuda SAKAMOTO, Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System, 2019, 42, 0387-3870, 10.3836/tjm/1502179295
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    5. Takashi Okuda Sakamoto, Hopf bifurcation in a reaction–diffusion system with conservation of mass, 2013, 26, 0951-7715, 2027, 10.1088/0951-7715/26/7/2027
    6. Hirofumi Izuhara, Shunusuke Kobayashi, An instability framework of Hopf–Turing–Turing singularity in 2-component reaction–diffusion systems, 2024, 0916-7005, 10.1007/s13160-024-00668-0
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  • © 2012 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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