Self-Replication, Self-Destruction, and Spatio-Temporal Chaos in the Gray-Scott Model

Yasumasa Nishiura¹ and Daishin Ueyama²

¹Laboratory of Nonlinear Studies and Computations, Research Institute for Electronic Science, Hokkaido University, Kita-ku, Sapporo 060-0812, Japan
²Department of Mathematical and Life Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan
E-mail: nishiura@aurora.es.hokudai.ac.jp, daishin@math.sci.hiroshima-u.ac.jp

(Received April 19, 2000; Accepted May 19, 2000)

Keywords: Spatio-Temporal Chaos, Self-Replicating Pattern, Reaction Diffusion System, Wave-Splitting

Abstract. A new geometrical understanding for the spatio-temporal chaos arising in the Gray-Scott model is presented. This is based on the interrelationship of global bifurcating branches of stationary patterns with respect to the supplying and removal rates contained in the model, especially their locations of saddle-node points and the Hopf bifurcation point of a constant state play a key role. It is possible to clarify the spatial structure of intermittent type of behavior by taking this point of view. At the onset point there exists a generalized heteroclinic cycle on the whole line and spatio-temporal chaos emerges by unfolding this cycle.