American Institute of Mathematical Sciences

December  2012, 7(4): 893-926. doi: 10.3934/nhm.2012.7.893

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, Japan 2 Meteorological college, Kashiwa, 277-0852, Japan

Received  January 2012 Revised  July 2012 Published  December 2012

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. Networks & Heterogeneous Media, 2012, 7 (4) : 893-926. doi: 10.3934/nhm.2012.7.893
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