Controlling Chaos: Theoretical and Practical Methods in by Tomasz Kapitaniak

By Tomasz Kapitaniak

Greater than 20 years of extensive stories on non-linear dynamics have raised questions about the sensible functions of chaos. One attainable resolution is to regulate chaotic habit in a predictable approach. This publication, oneof the 1st at the topic, explores the tips at the back of controlling chaos.
Controlling Chaos explains, utilizing uncomplicated examples, either the mathematical idea and experimental effects used to use chaotic dynamics to genuine engineering platforms. Chuas circuit is used to illustrate in the course of the publication because it could be simply built within the laboratory and numerically modeled. using this instance permits readers to check the theories provided. The textual content is thoroughly balanced among conception and functions to supply an in-depth exam of the suggestions in the back of the complicated rules offered. within the ultimate part, Kapitaniak brings jointly chosen reprinted papers that have had an important impact at the improvement of this swiftly starting to be interdisciplinary box. Controlling Chaos is key examining for graduates, researchers, and scholars wishing to be on the vanguard of this fascinating new department of science.
* makes use of effortless examples that are repeated by way of the reader either experimentally and numerically
* the 1st e-book to provide uncomplicated equipment of controlling chaos
* comprises reprinted papers representing primary contributions to the field
* Discusses implementation of chaos controlling basics as utilized to sensible difficulties

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Additional resources for Controlling Chaos: Theoretical and Practical Methods in Non-linear Dynamics

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E. 14d-f)) is on attractor Al. 12(a and b)). 12(b) is shown. Thus synchronization is normal (x = y, x = [X,,Yl,Zl] ~, y - [X2,Y2,Z2]T) in coupled Lorenz systems having identical values of r. When the values of r,, r2 are different, synchronization, by definition, cannot occur. 13(a), for a range of r,, r2 a practical synchronization takes place. 0. For Synchronization of chaos 300 250 Zl 200 150 100 -50 -25 0 xl 25 50 -25 0 xl 25 50 (a) 300 250 -72 200 150 100 -50 (b) 300 I . . I . . 14); (a)(b) attractors for r - 211, (c) s y m m e t r i c a l attractor for r - 219.

W. B. (1980) Chaotic motion in nonlinear feedback systems. IEEE Trans. Circ. , 27, 990-997. R. D. (1992) On dynamic control of chaos: a study with reference to a reacting system. Phys. Lett. A, 166, 197-204. A. (1994) Communications using chaotic frequency modulation. Int. J. Bifurcat. Chaos, 4, 427-440. Berreby, D. (1993) Chaos hits Wall Street. Discover, March, 76-84. , Derozier, D. and Glorieux, D. (1992) Controlling laser chaos. Proceedings of Nonlinear Dynamics in Optical Systems, Alpbach, Austria, June, 22-26.

1. We assume that some state variables of both systems A and B can be measured. Let us say that we can measure signal x~(t) from the system A and signal y,(t) from B where i e { 1, 2 . . , n}. 1) as negative feedback. 1 Synchronization by continous control. The parameter K > 0 is an experimentally adjustable weight of the perturbation and we shall discuss its selection later. Experimental realization of such feedback presents no difficulties for many practical systems. 4b). When synchronization is achieved F(t) becomes zero, so the chaotic systems A and B become practically uncoupled.

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