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A nested sequence of transitions for collision dynamics in dissipative systems
by Masaaki Yadome   Kei-Ichi Ueda   Takashi Teramoto   Masaharu Nagayama   Yasumasa Nishiura

Vol. 3 No. 4 (2008) P.585~P.601

ABSTRACT

We study the dynamics of head-on collisions of traveling pulses for a three-component reaction diffusion system. A variety of outputs with large deformation such as annihilation, repulsion, and fusion are observed after collision, however it remains open for a long time that what kind of mathematical structure controls the input-output relation at collision point. A series of works [18, 19, 20, 24] clarify some aspect of scattering dynamics that a network of unstable patterns called $\it scattors$ forms a backbone of the traffic control of input-output relations. Namely the unstable manifolds of those scattors constitute a network and complicated deformation processes and their transitions are controlled by rewiring those connections depending on parameters. In this article, by employing a three-component reaction diffusion system, we numerically show that there occurs a nested sequence of outputs among annihilation, repulsion, and fusion as parameters are varied in an appropriate way. It turns out that there exists a time-periodic unstable solution that plays a role of scattor and two heteroclinic connections are detected between the unstable periodic solution and other unstable stationary scattors which are responsible for the nested output of periodic type.

KEYWORDS
Reaction-diffusion systems, pattern formation, pulse dynamics, bifurcation

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 37M05, 37M20, 65P30

MILESTONES