By Tomasz Kapitaniak
Greater than 20 years of extensive experiences on non-linear dynamics have raised questions about the sensible functions of chaos. One attainable solution is to manage chaotic habit in a predictable approach. This publication, oneof the 1st at the topic, explores the guidelines in the back of controlling chaos. Controlling Chaos explains, utilizing easy examples, either the mathematical thought and experimental effects used to use chaotic dynamics to genuine engineering structures. Chuas circuit is used as an instance through the booklet because it could be simply developed within the laboratory and numerically modeled. using this instance permits readers to check the theories awarded. The textual content is thoroughly balanced among conception and functions to supply an in-depth exam of the strategies in the back of the advanced rules awarded. within the ultimate part, Kapitaniak brings jointly chosen reprinted papers that have had an important impression at the improvement of this speedily turning out to be interdisciplinary box. Controlling Chaos is vital analyzing for graduates, researchers, and scholars wishing to be on the vanguard of this interesting new department of technology. * makes use of effortless examples which might be repeated by means of the reader either experimentally and numerically* the 1st publication to provide easy tools of controlling chaos* contains reprinted papers representing basic contributions to the sphere* Discusses implementation of chaos controlling basics as utilized to functional difficulties
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Additional info for Controlling chaos : theoretical and practical methods in non- linear dynamics
Sample text
16(b). 0 have been taken. 18), as different behavior is obtained when control parameter is increased from when this parameter is decreased. This poses the question whether or not it is possible to desynchronize the systems again, and by what means. It is clear that any further parameter changes, in which the a = ot values of both attractors change simultaneously, will fail to desynchronize two systems. 6 (a) I 32 f,69 . . 2 x I i I I I ! 15 Evolution of coupled - 7; (b) unclosed curve c~- 8. 2 Lissajou figure a, = ~,, leaving the other at a 2 " - 0~, the two systems must have totally different attractors, B and A2, say, and hence are desynchronized.
15) is not initially fulfilled. 14 Necessary condition for synchronization of two chaotic systems. 9) will not be fulfilled) and in the x , - y,, i = 1, 2, . . , n plots we observe unclosed curves instead of a straight line. In this section we describe a simple method which allows us to obtain synchronization of periodic and chaotic trajectories evolving on different coexisting attractors A, and A 2 which are close to the single chaotic attractor B. Let us assume that the 'one attractor' a, and 'coexisting attractors' a: values of a control parameter a are close together.
6) is allowed in order to achieve synchronization. Knowing the equations of chaotic systems A and B we can easily check the above condition by direct computation of the Lyapunov exponents. 4 Numerical-vc,~ versus vc, chronized state. 6) will evolve on the same manifold on which both chaotic systems evolve and this is why synchronization can be obtained. When it is not fulfilled, the coupled system will evolve on a higher-dimensional manifold on which hyperchaotic attractor exists, and according to De Sousa et al.