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 purposes of chaos. One attainable solution is to regulate chaotic habit in a predictable method. This booklet, oneof the 1st at the topic, explores the guidelines at the back of controlling chaos.
Controlling Chaos explains, utilizing uncomplicated examples, either the mathematical thought and experimental effects used to use chaotic dynamics to actual engineering structures. Chuas circuit is used as an instance through the booklet because it will be simply built within the laboratory and numerically modeled. using this instance permits readers to check the theories offered. The textual content is punctiliously balanced among conception and functions to supply an in-depth exam of the thoughts at the back of the advanced principles provided. within the ultimate part, Kapitaniak brings jointly chosen reprinted papers that have had an important influence at the improvement of this speedily turning out to be interdisciplinary box. Controlling Chaos is vital examining for graduates, researchers, and scholars wishing to be on the leading edge of this intriguing new department of science.
* makes use of effortless examples which might be repeated by means of the reader either experimentally and numerically
* the 1st e-book to give easy equipment of controlling chaos
* comprises reprinted papers representing basic contributions to the field
* Discusses implementation of chaos controlling basics as utilized to useful difficulties

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

Sample text

As may be seen for K, - 22, when the rotational velocity is sufficiently high (m > co,) the self-excited vibrations (of conical modes - thin solid lines) appear for any value of C~, which lead to the damage of the beatings. We may avoid this danger, changing the stiffness coefficient K, in the time 2F. " - Kpl Kpl ~ [-~ Cp Kp2~T r . 7 Rotor system supported in two gas beatings. o- ~ t : ' ~ e ..... Conical . . . o,8. e ~-14 .... -' ,e" 9 9 ~ Stable Unstable . ~ e "~ (~ 4 ~ ~ ~ 9 e9 -.. . . (.......

3 Experimental vc,~ versus vc,~2~plots of unidirectionally-coupled Chua's circuits; (a) synchronized state, (b) unsynchronized state. , 1993). , 1993) feeding back only one state variable does not always result in a successful synchronization. One can easily show that synchronization can be achieved only if the number of positive Lyapunov exponents of the 'composite' coupled system is equal to the number of positive Lyapunov exponents of the component system. 6) is allowed in order to achieve synchronization.

12) where 6a is a vector describing differences in parameter values of both chaotic systems. 9) is fulfilled, so we have to replace it by weaker relation: lim IIe(t)ll < e. 13) where e is a vector of small parameters (e, ~ 1, i = 1,2, . . , n). 12) are practically (or noisily) synchronized. 10. 10). 4b) evolve on the same manifold on which both chaotic systems evolve (phase space is reduced to subspace x = y) and that is why synchronization can be obtained. When it is not fulfilled the coupled systems evolve on higher-dimensional manifold on which a hyperchaotic attractor exists.

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