By Ahmad Taher Azar

The publication stories at the most up-to-date advances in and purposes of chaos concept and clever regulate. Written by means of eminent scientists and energetic researchers and utilizing a transparent, matter-of-fact type, it covers complex theories, equipment, and functions in numerous learn components, and explains key suggestions in modeling, research, and regulate of chaotic and hyperchaotic structures. themes comprise fractional chaotic platforms, chaos keep an eye on, chaos synchronization, memristors, jerk circuits, chaotic structures with hidden attractors, mechanical and organic chaos, and circuit recognition of chaotic platforms. The booklet extra covers fuzzy good judgment controllers, evolutionary algorithms, swarm intelligence, and petri nets between different issues. not just does it give you the readers with chaos basics and clever control-based algorithms; it additionally discusses key purposes of chaos in addition to multidisciplinary strategies constructed through clever keep an eye on. The e-book is a well timed and complete reference consultant for graduate scholars, researchers, and practitioners within the components of chaos concept and clever control.

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**Sample text**

24) becomes 1 (25) iυ = − R3 which represents the i − υ characteristic of the negative resistor. If the voltage takes large values the slope of the i − υ characteristic is positive, since the op amp saturates. For example, for the positive saturation (υ0 = E sat ) the Eq. (21) can be written as i= υ − E Sat R1 (26) which now represents a i − υ characteristic which is translated with respect to the origin and has a positive slope. The breakpoint υ = E 1 of the whole i − υ characteristic of the nonlinear resistor can be calculated by substituting υ = E 1 in Eq.

Proof Here ea (t) and eb (t) are the parameter estimation errors given as ea (t) = a − A (t) eb (t) = b − B (t) . (17) Differentiating (17) with respect to t, we obtain e˙a (t) = − A˙ (t) e˙b (t) = − B˙ (t) . (18) Substituting adaptive control law (15) into (14), the closed-loop error dynamics is determined as ⎧ e˙x = −k x ex ⎪ ⎪ ⎨ e˙ y = − (a − A (t)) (x1 W (ϕ1 ) + x2 W (ϕ2 )) − 2 (b − B (t)) − k y e y (19) e˙z = −k z ez ⎪ ⎪ ⎩ e˙ϕ = −kϕ eϕ Dynamics, Synchronization and SPICE Implementation … 43 Then substituting (17) into (19), we have ⎧ e˙x = −k x ex ⎪ ⎪ ⎨ e˙ y = −ea (x1 W (ϕ1 ) + x2 W (ϕ2 )) − 2eb − k y e y e˙z = −k z ez ⎪ ⎪ ⎩ e˙ϕ = −kϕ eϕ (20) We consider the Lyapunov function as V (t) = V ex , e y , ez , eϕ , ea , eb = 21 ex2 + e2y + ez2 + eϕ2 + ea2 + eb2 .

EU result for linear equations of the form R D α y(x) − λy(x) = f (x), (n − 1 ≤ Re(α) < n), (1) where R D α is Riemann-Lioville fractional derivative is derived by Barret [47]. Al-Bassam [48] used method of successive approximation to derive EU result for nonlinear problem R D α y(x) = f (x, y(x)), (0 < α ≤ 1). (2) Delbosco and Rodino [49] derived EU results for nonlinear equations using Schauder’s fixed point theorem. Cauchy problems of the form (2) with Caputo derivative are investigated by Diethelm and Ford [50].