Oscillatory dynamics in a reaction-diffusion system in the presence of  0:1:2 resonance

Toshiyuki Ogawa; Takashi Okuda; Toshiyuki Ogawa; Takashi Okuda

doi:10.3934/nhm.2012.7.893

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

Toshiyuki Ogawa ^{1
,},
Takashi Okuda ^{2
,}

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.
- Hopf bifurcation,
- triply degenerate point,
- 0:1:2 resonance,
- normal form,
- chaotic dynamics
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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Abstract

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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