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Controlling Chaos: Suppression, Synchronization and by Huaguang Zhang

By Huaguang Zhang

Controlling Chaos deals its reader an intensive collection of ideas to accomplish 3 ambitions: the suppression, synchronization and iteration of chaos, every one of that's the focal point of a separate a part of the publication. The textual content bargains with the well known Lorenz, Rössler and Hénon attractors and the Chua circuit, and with much less celebrated novel structures. Modeling of chaos is entire utilizing distinction equations and usual and time-delayed differential equations. The equipment directed at controlling chaos enjoy the impact of complicated nonlinear keep watch over concept: inverse optimum regulate is used for stabilization; distinctive linearization for synchronization; and impulsive keep an eye on for chaotification. significantly, a fusion of chaos and fuzzy structures theories is hired, with the Takagi–Sugeno version and the authors’ personal fuzzy hyperbolic version used in the modeling and keep an eye on of chaotic structures. Time-delayed structures also are studied with many synchronization tools being explored. the entire effects offered are normal for a large type of chaotic systems.

This monograph is self-contained with introductory fabric delivering a assessment of the background of chaos regulate and the required mathematical preliminaries for operating with dynamical systems.

Controlling Chaos may be of curiosity to lecturers from electric, structures, mechanical and chemical engineering backgrounds operating up to speed thought with regards to nonlinear dynamical and chaotic structures and to graduate scholars of chaos control.

The Communications and keep watch over Engineering sequence studies significant technological advances that have strength for nice effect within the fields of conversation and keep watch over. It displays study in commercial and educational associations around the globe in order that the readership can make the most new chances as they develop into available.

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Additional info for Controlling Chaos: Suppression, Synchronization and Chaotification

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No sequences of points approach the equilibrium for n → ±∞ except for those originating from points on the eigenvectors associated with κ2 . Again, if κ1 , κ2 > 0 orbits move monotonically (see Fig. 11 (b)); otherwise they oscillate improperly (see Fig. 11 (d)). The stable and unstable eigenspaces are one dimensional. Case 2: Δ < 0. In this case, det(B) > 0. Eigenvalues are a complex conjugate pair given by (κ1 , κ2 ) = (κ , κ¯) = σ ± iθ and solutions are sequences of points situated on a spiral whose amplitude increases or decreases in time according to the factor rn , where r = |σ ± iθ | = σ 2 + θ 2 = det(B) is the modulus of the complex eigenvalue pair.

3 Poincar´e Map A Poincar´e map is a classical technique for analyzing dynamical systems. It replaces the flow of an nth-order continuous-time system with an (n − 1)st-order discrete-time system. The definition of the Poincar´e map ensures that the limit sets of the discrete-time system correspond to the limit sets of the underlying flow. The Poincar´e map’s usefulness lies in the reduction of order and the fact that it bridges the gap between continuous-time and discrete-time systems. The definitions of a Poincar´e map are different for autonomous systems and nonautonomous systems.

We can convert it into an (n + 1)st-order autonomous system by appending an extra state θ := 2π t/T. 13) θ˙ = 2π /T, θ (t0 ) = 2π t0 /T. 13) is periodic in θ with period 2π . Hence, the planes θ = 0 and θ = 2π may be identified and the state space transformed from the Euclidean space Rn+1 to the cylindrical space Rn × S1 , where S1 is the unit circle. 14) where the modulo function restricts to 0 ≤ θ < 2π . Consider the n-dimensional hyperplane Σ ∈ Rn × S1 defined by Σ := {(x, θ ) ∈ Rn × S1 : θ = θ0 }.

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