Numerical Studies of Pattern Formation and Lyapunov Exponents in Chaotic Reaction–Diffusion Systems

Hiroyuki NAGASHIMA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.11   pp.1806-1810
Publication Date: 1994/11/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)
Category: Chaos and Related Topics
pattern formation,  chaos,  Lyapunov exponents,  reaction–diffusion system,  

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Numerical studies of reaction–diffusion systems which consist of chaotic oscillators are carried out. The Rössler oscillators are used, which are arranged two–dimensionally and coupled by diffusion. Pacemakers where the average periods of the oscillators are artificially changed are set to produce target patterns. It is found that target patterns emerge from pacemakers and grow up as if they were in a regular oscillatory medium. The wavelength of the pattern can be varied and controlled by changing the parameters (size and frequency) of the pacemaker. The behavior of the coupled system depends on the size of the system and the strength of the pacemaker. When the system size is large, the Poincar return maps show that the behavior of the coupled system is not simple and the orbit falls into a high–dimensional attractor, while for a small system the attractor is rather simple and a one–dimensional map is obtained. Moreover, for appropriate strength of pacemakers and for certain sizes of the systems the oscillations become periodic. It is also found that the largest and local Lyapunov exponents of the system are positive and these values are uniformly distributed over the pattern. The values of the exponents are smaller than that of the uncoupled Rössler oscillator; this is due to the fact that the diffusion reduces the exponents and modifies the form of the attractor. We conclude that the large scale patterns can stably exist in the chaotic medium.