Fuzzy Chaotic Systems: Modeling, Control, and Applications by Zhong Li
By Zhong Li
"Fuzzy Chaotic structures" offers unique heuristic study achievements and insightful rules at the interactions or intrinsic relationships among fuzzy good judgment and chaos conception. It provides the elemental options of fuzzy common sense and fuzzy keep watch over, chaos idea and chaos keep an eye on, in addition to thedefinition of chaos at the metric house of fuzzy units. This monograph discusses and illustrates fuzzy modeling and fuzzy keep watch over of chaotic platforms, synchronization, anti-control of chaos, clever electronic remodel, spatiotemporal chaos and synchronization in complicated fuzzy structures; in addition to a realistic software instance of fuzzy-chaos-based cryptography. Like different excellent books, this booklet could elevate extra questions than it will probably supply solutions. It as a result generates an outstanding capability to draw extra recognition to mix fuzzy platforms with chaos concept and comprises very important seeds for destiny medical learn and engineering applications.
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Extra resources for Fuzzy Chaotic Systems: Modeling, Control, and Applications
The OGY method is very ﬂexible to stabilize diﬀerent periodic orbits at diﬀerent times. To switch from stabilizing a period-2 orbit to, say, stabilizing the ﬁxed point in order to achieve a better system performance at a later time, we just need to turn oﬀ the parameter control with respect to the period-2 orbit. Thus, without control, the trajectory will diverge from the period-2 orbit exponentially, and resume to evolve with the parameter value µ0 . Later, 48 3 Chaos and Chaos Control the trajectory enters a small neighborhood of the ﬁxed point.
A fundamental reason lies in that chaotic systems with complex nonlinear dynamical behaviors and extreme sensitivity to initial conditions are deterministic by their very nature, and follow a set of deterministic rules that can be used in designing feedback controllers for diﬀerent purposes . 34) where x(t) is the system state vector, f a continuous or smooth nonlinear vector-valued function satisfying some necessary conditions, such as the wellposedness of the controlled system and uniqueness of the system trajectory under any admissible control u(t), for any initial position x0 in a region of interest, t0 ≤ t < ∞.
Fig. 4. 13) √ has a ﬁxed point x1 = 0 at µ0 = 0, and ﬁxed points x2 = ± µ at µ > 0, √ √ where x2 = µ is stable and x∗2 = − µ is unstable for µ > 0. Hence, by the nature of the bifurcation point (x, µ) = (0, 0) in the x − µ plane, as shown in Fig. 5, it is called a saddle-node bifurcation. x ✻ x2 = µ µ 0 ✲ Fig. 5. 3 (Pitchfork bifurcation). 14) √ has a ﬁxed point x1 = 0 at µ0 = 0, and ﬁxed point x2 = ± µ at µ > 0. Since √ x0 = 0 is unstable for µ > µ0 = 0 and stable for µ < µ0 = 0, and x2 = ± µ is stable for µ > 0 due to the fact that the Jacobian J = −2µ, this case is called a pitchfork bifurcation, as shown in Fig.