Chaos in Automatic Control by Wilfrid Perruquetti, Jean-Pierre Barbot

By Wilfrid Perruquetti, Jean-Pierre Barbot

Chaotic habit arises in various keep an eye on settings. every so often, it truly is worthy to take away this habit; in others, introducing or benefiting from the present chaotic elements might be precious for instance in cryptography. Chaos in computerized regulate surveys the most recent equipment for putting, profiting from, or removal chaos in quite a few purposes. This e-book provides the theoretical and pedagogical foundation of chaos up to speed structures besides new ideas and up to date advancements within the box. provided in 3 elements, the publication examines open-loop research, closed-loop keep watch over, and purposes of chaos on top of things platforms. the 1st part builds a heritage within the arithmetic of normal differential and distinction equations on which the rest of the publication relies. It comprises an introductory bankruptcy by way of Christian Mira, a pioneer in chaos learn. the following part explores suggestions to difficulties bobbing up in remark and keep watch over of closed-loop chaotic keep an eye on structures. those contain model-independent regulate tools, concepts equivalent to H-infinity and sliding modes, polytopic observers, common types utilizing homogeneous adjustments, and observability basic kinds. the ultimate part explores purposes in instant transmission, optics, energy electronics, and cryptography. Chaos in automated keep an eye on distills the most recent considering in chaos whereas pertaining to it to the latest advancements and functions up to the mark. It serves as a platform for constructing extra powerful, self sustaining, clever, and adaptive structures.

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Extra resources for Chaos in Automatic Control

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The case Si (X, b ) = e±jϕ , i = 1 or 2, j2 = −1, corresponds to a Neimark bifurcation (frequently and erroneously attributed to Hopf ). In the simplest case, for example, when crosses through b a stable (resp. unstable) focus point becomes unstable (resp. stable) and gives rise to a stable (resp. unstable) invariant closed curve (γ ). The corresponding bifurcation curve j ( k ) in the parameter plane is called a Neimark curve. Fold, flip, pitchfork, and Neimark bifurcation curves are given in a parametric form [the vector X being the parameter of the parametric form, Si (X, ) being one of the two multipliers of the cycle (k, j) considered here] by the following relations: X = T k (X, ), X = T k (X, ), for r < k, dim X = 2 Si (X, ) = +1, i = 1 or 2, for fold and pitchfork curves Si (X, ) = −1, i = 1 or 2, for flip curves Si (X, ) = e±jϕ , i = 1 or 2, j2 = −1, for Neimark curves The Neimark bifurcation may gives rise to several situations when ϕ is commensurable with 2π .

Codimension-2 points correspond to complex communications between the sheets. Therefore, the association of fold and flip curves in the neighborhood of a fold cusp leads to the definition of three fundamental communication types: the crossroad area (CRA), the saddle area (SAA), and the spring area (SPA) [108, 114]. Other types of singularities, with the corresponding three-dimensional representation of the sheets, are described in Carcassès et al. [29], Mira et al. [115, 119], Mira [110], Allam and Mira [8], and Mira and Qriouet [118].

For example, a stable (k; j)-node cycle (0 < S1 < 1, 0 < S2 < 1) gives rise to a (k; j)-saddle cycle (S1 < 1, S2 > 1) with two stable (k; j)-node cycles (0 < S1 < 1, 0 < S2 < 1), with all these cycles merging for a parameter point on the pitchfork curve. The case Si (X, b ) = e±jϕ , i = 1 or 2, j2 = −1, corresponds to a Neimark bifurcation (frequently and erroneously attributed to Hopf ). In the simplest case, for example, when crosses through b a stable (resp. unstable) focus point becomes unstable (resp.

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