By Wilfrid Perruquetti, Jean-Pierre Barbot
Chaotic habit arises in various keep an eye on settings. on occasion, it really is invaluable to take away this habit; in others, introducing or benefiting from the prevailing chaotic elements could be priceless for instance in cryptography. Chaos in automated regulate surveys the most recent tools for placing, making the most of, or elimination chaos in a number of functions. This publication provides the theoretical and pedagogical foundation of chaos on top of things structures besides new techniques and up to date advancements within the box. provided in 3 components, the e-book examines open-loop research, closed-loop regulate, and purposes of chaos on top of things platforms. the 1st part builds a historical past within the arithmetic of standard differential and distinction equations on which the rest of the ebook relies. It contains an introductory bankruptcy via Christian Mira, a pioneer in chaos study. the subsequent part explores strategies to difficulties bobbing up in commentary and keep watch over of closed-loop chaotic keep watch over structures. those contain model-independent keep watch over equipment, ideas resembling H-infinity and sliding modes, polytopic observers, general kinds utilizing homogeneous alterations, and observability basic kinds. the ultimate part explores purposes in instant transmission, optics, strength electronics, and cryptography. Chaos in automated keep watch over distills the most recent considering in chaos whereas bearing on it to the latest advancements and functions up to the mark. It serves as a platform for constructing extra powerful, independent, clever, and adaptive structures.
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Additional info for Chaos in Automatic Control
2 Chaos and Unpredictability . . . . . . . . . . . . 3 Generalities on Discrete Models . . . . . . . . . . . 1 Different Forms of Models . . . . . . . . . . 2 Maps Obtained from an ODE by a Poincaré Section . 4 Singularities and Bifurcations Common to Invertible and Noninvertible Maps . . . . . . . . . . . . . . . 1 Singularities and Bifurcations . . . . . . . . . 2 Bifurcation Sets: Normal Forms of Exceptional Critical Cases .
The first corresponds to qualitative methods [9–11]. The “strategy” of these methods can be defined noting that the solutions of equations of nonlinear dynamic systems are in general nonclassical, nontabulated, transcendental functions of mathematical analysis, which are very complex. This strategy is of the same type as the one used for the characterization of a complex variable function by its singularities: zeros, poles, essential singularities. Here, the complex transcendental functions are defined by the singularities of continuous (resp.
A rank-q critical set CMq−1 is given by the rank-q image CMq−1 = T q (CM), CM0 ≡ CM. If dim X = p = 1, CM is a rank-one critical point C. If dim X = p = 2, CM is a rank-one critical curve LC. Such new singularities play a fundamental role in the attractors and basins structure and in their bifurcations. It is the case of “contact bifurcations,” resulting from the meeting of two singularities of different nature: an invariant manifold (or set) by T or T −1 with a critical set. This situation generally gives rise to global bifurcations, which may be related to homoclinic and heteroclinic bifurcations [51, 53, 98, 121].