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By Zhong Li

"Fuzzy Chaotic structures" presents unique heuristic learn achievements and insightful rules at the interactions or intrinsic relationships among fuzzy common sense and chaos concept. It offers the elemental thoughts of fuzzy common sense and fuzzy regulate, chaos idea and chaos keep watch over, in addition to thedefinition of chaos at the metric area of fuzzy units. This monograph discusses and illustrates fuzzy modeling and fuzzy keep watch over of chaotic structures, synchronization, anti-control of chaos, clever electronic remodel, spatiotemporal chaos and synchronization in complicated fuzzy platforms; in addition to a pragmatic program instance of fuzzy-chaos-based cryptography. Like different excellent books, this publication may possibly bring up extra questions than it will possibly offer solutions. It accordingly generates a superb strength to draw extra consciousness to mix fuzzy platforms with chaos conception and includes very important seeds for destiny clinical study and engineering applications.

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The fuzzifier maps crisp points in U to fuzzy sets in U , and the defuzzifier maps fuzzy sets in V to crisp points in V . 5 Fuzzifier A fuzzifier performs a map from a crisp point x = (x1 , . . , xn )T into a fuzzy set A in U [59]. , µA (x ) = 1 for x = x and µA (x ) = 0 for all other x ∈ U with x = x. Nonsingleton fuzzifier : µA (x) = 1 and µA (x ) decreases from 1 as x moves away from x, for example, µA (x ) = exp − (x −x)σ2(x −x) , where σ 2 is a T parameter characterizing the shape of µA (x ). It seems that only the singleton fuzzifier has been applied in practice.

1 Introduction Although a unified, universally accepted, and rigorous mathematical definition of chaos is still not available in the scientific literature, various alternative, but closely related definitions of chaos have been proposed, among which those of Li-Yorke and Devaney seem to be the most popular ones. , the well-known “period three implies chaos” [129]. Consider a one-dimensional discrete dynamical system [19, 128]: xk+1 = f (xk ), k = 0, 1, 2, . . 1) where xk ∈ J (an interval) and f : J → J is a continuous map.

In the case of the logistic map, we have 1 λ(x0 ) = lim k→∞ k k−1 ln |µ − 2µxi | i=0 1 k→∞ k k−1 ln |1 − 2xi |. 19) i=0 Especially, note that since x = 0 is a fixed point, we have λ(0) = ln µ (and also λ(1) = ln µ). 7 shows the Lyapunov exponent for the logistic map as a function of µ. The Lyapunov exponent is independent on the initial value x0 . We can see that the values of µ for which λ becomes negative in Fig. 7 correspond to the regions of periodic behavior evident in Fig. 2. 1 is called period-doubling bifurcation, which states that as µ increases, a sequence of period-doubling bifurcations occurs.

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