Infinite dimensional relaxation oscillation in aggregation-growth systems

Pages: 1859 - 1887, Volume 17, Issue 6, September 2012

doi:10.3934/dcdsb.2012.17.1859       Abstract        References        Full Text (1391.6K)       Related Articles

Shin-Ichiro Ei - Institute of Mathematics for Industry, Kyusyu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan (email)
Hirofumi Izuhara - Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, 214-8571, Japan (email)
Masayasu Mimura - Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, 214-8571, 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.
Mathematics Subject Classification:  Primary: 35K57, 35B10; Secondary: 35R15.

Received: October 2011;      Revised: January 2012;      Published: May 2012.

 References