Bulletin of the Japan Society for Industrial and Applied Mathematics

In 1952, Turing showed that diffusion does possibly enhance spatial inhomogeneities in some situation, by using simple reaction-diffusion equations. This paradoxical argument demonstrates that diffusion generates spatio-temporal patterns, which is mathematically supported by the stability theory of differential equations. In many biological, chemical and physical systems, we have encountered a variety of diffsuion-induced spatiotemporal patterns suggested by Turing. In this note, among these patterns, we concentrate ourselves on bacterial colony patterns arising in biological systems. We discuss on the diversity of colony patterns observed in experiments, by using a reaction diffusion equation model which is proposed on the idea of Turing's paradoxical argument. Furthermore, we apply this idea to understanding of spatio-temporal patterns arising in more general consumer-resource systems. As an example, we discuss combustion in micro-gravity circumstance from pattern formation viewpoints.

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We formulate a theory of the aging phenomena in shape memory alloys. In our model equations, the order parameter of the martensitic transformation is coupled with a secondary variable which follows the evolution of the order parameter but is assumed to be an extremely slow variable. From this set of equations, we derive the interface (twin boundary) equation of motion. Solving the interface equation, the stress-strain relation is obtained by taking account of the aging effect. The results are found to be consistent, at least qualitatively, with experiments.

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Existence and dynamics of chaotic pulses on 1 D lattice are discussed. Traveling pulses arise typically in reaction diffusion systems like FitzHugh-Nagumo equations. Such pulses annihilate when they collide each other. A new type of traveling pulse has been found recently in many systems where pulses bounce off like elastic balls. We consider the behavior of such a localized pattern on 1 D lattice, i. e., an infinite system of ODEs with nearest interaction of diffusive type. Besides the usual standing and traveling pulses, a new type of localized pattern, which moves chaotically on a lattice, was found numerically. Employing the strength of diffusive interaction as a bifurcation parameter, there appear two types of route from standing pulse to chaotic pulse; period-doubling and intermittent type I. If two chaotic pulses collide with an appropriate timing, it forms a periodic oscillating pulse called mollecule. Interaction among many chaotic pulses is also studied numerically.

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In the present article, we study the existing condition of stable helically propagating combustion wave and transition process of wave patterns. Through our three-dimensional numerical simulation, we get the following results: (1) When physical parameters are fixed so that a combustion wave of steady-state mode is stable in the one-dimensional problem, the planar wave of steady-state mode is stable also in the cylindrical domain. (2) Set physical parameters so that a pulsating wave exists stably for the one-dimensional problem. Then, the planar pulsating wave is still stable in the cylindrical domain if the radius is small while a helical wave takes the place of the pulsating wave if the radius becomes larger. Moreover, we obtain new insights about propagation patterns in the interior of cylindrical domain, differences in propagation patterns between the two-and three-dimensional problems and dependence of propagation patterns on the effective activation energy. The article describes also a finite difference approximation for the Laplacian operator in the cylindrical coordinate.

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