Infinite dimensional relaxation oscillation in aggregation-growth systems
Shin-Ichiro Ei - Institute of Mathematics for Industry, Kyusyu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan (email)
Abstract: Two types of aggregation systems with Fisher-KPP growth are proposed. One is described by a normal reaction-diffusion system, and the other is described by a cross-diffusion system. If the growth effect is dominant, a spatially constant equilibrium solution is stable. When the growth effect becomes weaker and the aggregation effect become dominant, the solution is destabilized so that spatially non-constant equilibrium solutions, which exhibit Turing's patterns, appear. When the growth effect weakens further, the spatially non-constant equilibrium solutions are destabilized through Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when the growth effect is extremely weak, there appear spatio-temporal periodic solutions exhibiting infinite dimensional relaxation oscillation.
Keywords: Relaxation oscillation, aggregation-growth system, pattern formation, periodic solution, bifurcation.
Received: October 2011; Revised: January 2012; Published: May 2012.
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