This chapter introduces optimization algorithms for parameter extractions of three fractional-order circuits that model bioimpedance. The Cole-impedance model is investigated; it is considered one of the most commonly used models providing the best fit with the measured data. Two new models are introduced: the fractional Hayden model and the fractional-order double-shell model. Both models are the generalization of their integer-order counterpart. These fractional-order models provide an improved description of observed bioimpedance behavior. New metaheuristic optimization algorithms for extracting the impedance parameters of these models are investigated. The proposed algorithms inspired by nature are known as the Flower Pollination Algorithm, the Grey Wolf Optimizer, the Moth-flame Optimizer, the Whale Optimization Algorithm, and the Grasshopper Optimization Algorithm. These algorithms are tested over sets of simulated and experimental data. Their results are compared with a conventional fitting algorithm (the nonlinear least square) in aspects of speed, accuracy, and precision. © 2018 Elsevier Inc. All rights reserved.
FPGA implementation of integer/fractional chaotic systems
Chaotic systems have remarkable importance in capturing some complex features of the physical process. Recently, fractional calculus becomes a vigorous tool in characterizing the dynamics of complex systems. The fractional-order chaotic systems increase the chaotic behavior in new dimensions and add extra degrees of freedom, which increase system controllability. In this chapter, FPGA implementation of different integer and fractional-order chaotic systems is presented. The investigated integer-order systems include Chua double scroll chaotic system and the modified Chua N-scroll chaotic system. The investigated fractional-order systems include Chua, Yalcin et al., Ozuogos et al., and Tang et al., chaotic systems. These systems are implemented and simulated based on the Grunwald–Letnikov (GL) definition with different window sizes. The parameters effect, along with different GL window sizes is investigated where some interesting chaotic behaviors are obtained. The proposed FPGA implementation utilizes fewer resources and has high throughput. Experimental results are provided on a digital oscilloscope. © Springer Nature Switzerland AG 2020.
Generalized synchronization of different dimensional integer-order and fractional order chaotic systems
In this work different control schemes are proposed to study the problem of generalized synchronization (GS) between integer-order and fractionalorder chaotic systems with different dimensions. Based on Lyapunov stability theory of integer-order differential systems, fractional Lyapunov-based approach and nonlinear controllers, different criterions are derived to achieve generalized synchronization. The effectiveness of the proposed control schemes are verified by numerical examples and computer simulations. © Springer International Publishing AG 2017. All rights reserved.
Fractional-order oscillators
Fractional-order calculus is the branch of mathematics which deals with non-integerorder differentiation and integration. Fractional calculus has recently found its way to engineering applications; particularly electronic circuits with promising results showing the feasibility of fabricating fractional-order capacitors on silicon. Fractionalorder capacitors are lossy non-deal capacitors with an impedance given by Zc = (1/j?C)?, where C is the pseudo-capacitance and ? is its order (0 < ? ? 1). When these fractional-order capacitors are employed within an oscillator (sinusoidal or relaxation) circuit, this oscillator is called a fractional-order oscillator and is described by non-integer-order differential equations. Therefore, an oscillator of order 1.5 or 2.6 is possible to obtain. While the oscillation frequency in integer-order oscillators is related to their RC time constants, fractional-order oscillators have their oscillation frequencies also related to ?. This adds more design freedom and enables extremely high or extremely low oscillation frequencies even with large RC time constants. This chapter aims at reviewing the theory of designing fractional-order oscillators accompanied by several design examples. Experimental results are also shown. © The Institution of Engineering and Technology 2017. All rights reserved.
Nonlinear fractional order boundary-value problems with multiple solutions
It is well-known that discovering and then calculating all branches of solutions of fractional order nonlinear differential equations with boundary conditions can be difficult even by numerical methods. To overcome this difficulty, in this chapter two semianalytic methods are presented to predict and obtain multiple solutions of nonlinear boundary value problems. These methods are based on the homotopy analysis method (HAM) and Picard method namely, predictor HAM and controlled Picard method. The used techniques are capable of predicting and calculating all branches of the solutions simultaneously. Four problems are solved, three of them are practical problems which are generalized in fractional order domain to show the efficiency and importance of these methods. And the solutions are calculated by simple procedures without any need for special transformations or perturbation techniques. © 2018 Elsevier Inc. All rights reserved.
On the fractional order generalized discrete maps
Chaos theory describes the dynamical systems which exhibit unpredictable, yet deterministic, behavior. Chaotic systems have a remarkable importance in both modeling and information processing in many fields. Fractional calculus has also become a powerful tool in describing the dynamics of complex systems such as fractional order (FO) chaotic systems. The FO parameter adds extra degrees of freedom which increases the design flexibility and adds more control on the design. The extra parameters increase the chaotic range. This chapter provides a review of several generalized discrete time one-dimensional maps. The generalizations include a signed control parameter, scaling parameters, and shaping parameters. The properties of the generalized fractional logistic map are presented. The generalized fractional tent map is presented and its properties are studied and validated using numerical simulations. Various simulations are conducted including time series, bifurcation diagrams, and various chaotic properties against the system parameters and FO parameter. © 2018 Elsevier Inc. All rights reserved.
Control and synchronization of fractional-order chaotic systems
The chaotic dynamics of fractional-order systems and their applications in secure communication have gained the attention of many recent researches. Fractional-order systems provide extra degrees of freedom and control capability with integer-order differential equations as special cases. Synchronization is a necessary function in any communication system and is rather hard to be achieved for chaotic signals that are ideally aperiodic. This chapter provides a general scheme of control, switching and generalized synchronization of fractional-order chaotic systems. Several systems are used as examples for demonstrating the required mathematical analysis and simulation results validating it. The non-standard finite difference method, which is suitable for fractional-order chaotic systems, is used to solve each system and get the responses. Effect of the fractional-order parameter on the responses of the systems extended to fractional-order domain is considered. A control and switching synchronization technique is proposed that uses switching parameters to decide the role of each system as a master or slave. A generalized scheme for synchronizing a fractional-order chaotic system with another one or with a linear combination of two other fractional-order chaotic systems is presented. Static (timeindependent) and dynamic (time-dependent) synchronization, which could generate multiple scaled versions of the response, are discussed. © Springer International Publishing AG 2017. All rights reserved.
Memcapacitor: Modeling, analysis, and emulators
This chapter reviews the memcapacitor, mathematical representations of time-invariant, physical realizations, and mathematical models. Moreover, the nonlinear boundary effect of the memcapacitor under step, sinusoidal, and general periodic excitation responses are discussed with analytical, numerical, and circuit simulations for different examples. The general analyses of series and parallel connections of memcapacitors are introduced with many examples and circuit simulations. Finally a charge-controlled, memristor-less memcapacitor is introduced and validated through different cases. © 2015, Springer International Publishing Switzerland.
Meminductor: Modeling, analysis, and emulators
This chapter introduces the basic definition of meminductor and its mathematical representation of time-invariant system (Ideal, Generic, and Extended) with some examples. The mathematical model of meminductor and its response under different current excitations (step, sinusoidal, and periodic) are discussed with analytical, numerical, and circuit simulations. Different meminductor emulators are introduced with their mathematical modeling and numerical simulation, and verified using PSPICE simulations. © 2015, Springer International Publishing Switzerland.
Memristor mathematical models and emulators
This chapter introduces different generalized mathematical classes of memristors which can be categorized as: continuous symmetrical models (current and voltage controlled emulators), continuous nonsymmetrical model, switched-memristor model, and fractional-order model with some experimental results. Different emulators with experimental results are discussed based on CCII, discrete components, and MOS realizations. Different analytical expressions, numerical analyses, circuit simulations results as well as experimental results are provided for most of the previous models. © 2015, Springer International Publishing Switzerland.