By Professor Hazime Mori, Professor Dr. Yoshiki Kuramoto (auth.)

ISBN-10: 3642803768

ISBN-13: 9783642803765

ISBN-10: 3642803784

ISBN-13: 9783642803789

This monograph includes elements and provides an method of the physics of open nonequilibrium structures. half I derives the phenomena of dissipative constructions at the foundation of lowered evolution equations and comprises Bénard convection and Belousov-Zhabotinskii chemical reactions. half II discusses the physics and constructions of chaos. whereas proposing a building of the statistical physics of chaos, the authors unify the geometrical and statistical descriptions of dynamical structures. the form of chaotic attractors is characterised, as are the blending and diffusion of chaotic orbits and the fluctuation of power dissipation exhibited via chaotic systems.

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**Extra resources for Dissipative Structures and Chaos**

**Example text**

The first of these relations implies that w is the most rapidly varying and v is the most slowly varying among the three functions of time. For this reason, it should be possible to adiabatically eliminate w. 19b) by the resulting relation, w = bv / (u + a). 20a) dv/dt=u-v. 20b) 18 1. A Representative Example of Dissipative Structure The fundamentals of the theory concerning the reduction method based on this kind of adiabatic elimination are given in Chap. 5. 20b) is referred to as the Keener-Tyson (KT) model.

Thinking of w as w+, from the asymptotic form of this function, we find that the value of this integral is 271"i(1 _ q2). Thus we obtain the following relation for c: -iq c(q) 00 -l-'dq(1 _ q2) , ~ ~ 2W{JJ ded~I:l'r . Ld can be thought of as the defect's mobility. The direction of the motion of the defect relative to the roll axis depends on q. A pure climb results when qy = O. In this case, the t;-1] and x-y axes coincide, and q = qx' For q > 0 - in other words, when the distance between rolls is smaller than the critical wavelength Ac = 211"/ kc - the defect falls along a line parallel to the y-axis.

Physically, the above discussion implies the following. Defects are not composed of localized, rigid formations, but rather, they and their motion arise out of some soft, deformable structures. If e is small, the corresponding deformation is also small, but even in this case we cannot ignore the deformation. It is not possible for the Fourier modes from which some localized structure is constructed to adiabatically follow the slow motion of a defect, because it is always the case that even slower long-wavelength modes exist.

### Dissipative Structures and Chaos by Professor Hazime Mori, Professor Dr. Yoshiki Kuramoto (auth.)

by Daniel

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