Despite the complex nature of fractional calculus, it is still fairly possible to reduce this complexity by using integer-order approximation. Each integer-order approximation has its own trade-offs from the complexity, sensitivity, and accuracy points of view. In this chapter, two different fractional-order electronic circuits are studied: the Wien oscillator and the CCII-based KHN filter with two different fractional elements of orders ? and ?. The investigation is concerned with changes in the response of these two circuits under two approximations: Oustaloup and Matsuda. A detailed review of each approximation technique is provided as well as its design procedure. Oscillator and filter responses are simulated using MATLAB. Foster-I realization is used to implement the approximated Wien oscillator and filter transfer functions as circuits in order to simulate them in PSpice. The responses are compared to the exact solution to investigate which achieves the lowest error. For oscillators, the comparison is based on oscillation condition and oscillation frequency while for filters, the focus is on filter fundamental frequencies. This is a big issue in filter design: maximum or minimum frequency, right phase frequency, and half-power frequency. © 2018 Elsevier Inc. All rights reserved.
Survey on Two-Port Network-Based Fractional-Order Oscillators
This chapter merges the fractional calculus and two-port networks in oscillator design. The fractional-order elements ? and ? add extra degrees of freedom that increase the design flexibility and frequency band while providing control over the phase difference. A prototype of the fractional-order two-port network oscillators is introduced. It consists of a general two-port network and three impedances distributed as input, output, and a feedback impedance. Three different two-port network classifications are obtained according to the ground location. This chapter focuses on one of these classifications from which two derived prototypes can be extracted. The general analytical formulas of the oscillation frequency and condition as well as the phase difference are derived in terms of the transmission matrix parameter of a general two-port network. Different active building blocks are used to serve as a two-port network. Numerical, Spice simulations, and experimental results are given to validate the presented analysis. © 2018 Elsevier Inc. All rights reserved.
FPGA Implementation of Fractional-Order Chaotic Systems
This chapter introduces two FPGA implementations of the fractional-order operators: the Caputo and the Grünwald-Letnikov (GL) derivatives. First, the Caputo derivative is realized using nonuniform segmentation to reduce the size of the Look-Up Table. The Caputo implementation introduced can generate derivatives of previously defined functions only. Generic and complete hardware architecture of the GL operator is realized with different memory window sizes. The generic architecture is used as a block to implement several fractional-order chaotic systems. The investigated systems include Borah, Chen, Liu, Li, and Arneodo fractional-order chaotic systems. Different interesting attractors are realized under various parametric changes with distinct step sizes for different fractional orders. To verify the chaotic behavior of the generated attractors, the Maximum Lyapunov Exponent is calculated for each system at different parameter values. © 2018 Elsevier Inc. All rights reserved.
Fractional-Order Filter Design
One of the advantages of fractional order is the extra degree of freedom added by the fractional-order parameters, which enrich the analysis with more details in new dimensions. This chapter introduces factional-order conventional filters of orders ?, 2?, and 3?. The general transfer functions of continuous-time filters (low-pass, high-pass, and band-pass filters) to the noninteger-order (fractional-order) domain are investigated. Also, mathematical expressions for the maximum and minimum frequencies, the half power frequencies, and the right-phase frequencies are derived. In addition, the effect of the transfer function parameters on the filter poles and hence the stability is introduced. Numerical spice results are introduced to validate the theoretical findings. Several passive and active filters are studied to validate the concept. This chapter also investigates the effect of an inserted delay parameter on the filter main frequencies. Different filter responses are obtained from the general delayed transfer function. Two delay examples are investigated. © 2018 Elsevier Inc. All rights reserved.
Biologically Inspired Optimization Algorithms for Fractional-Order Bioimpedance Models Parameters Extraction
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.
Self-excited attractors in jerk systems: Overview and numerical investigation of chaos production
Chaos theory has attracted the interest of the scientific community because of its broad range of applications, such as in secure communications, cryptography or modeling multi-disciplinary phenomena. Continuous flows, which are expressed in terms of ordinary differential equations, can have numerous types of post transient solutions. Reporting when these systems of differential equations exhibit chaos represents a rich research field. A self-excited chaotic attractor can be detected through a numerical method in which a trajectory starting from a point on the unstable manifold in the neighborhood of an unstable equilibrium reaches an attractor and identifies it. Several simple systems based on jerk-equations and different types of nonlinearities were proposed in the literature. Mathematical analyses of equilibrium points and their stability were provided, as well as electrical circuit implementations of the proposed systems. The purpose of this chapter is double-fold. First, a survey of several self-excited dissipative chaotic attractors based on jerk-equations is provided. The main categories of the included systems are explained from the viewpoint of nonlinearity type and their properties are summarized. Second, maximum Lyapunov exponent values are explored versus the different parameters to identify the presence of chaos in some ranges of the parameters. © 2018, Springer International Publishing AG.
Chaotic properties of various types of hidden attractors in integer and fractional order domains
Nonlinear dynamical systems with chaotic attractors have many engineering applications such as dynamical models or pseudo-random number generators. Discovering systems with hidden attractors has recently received considerable attention because they can lead to unexpected responses to perturbations. In this chapter, several recent examples of hidden attractors, which are classified into several categories from two different viewpoints, are reviewed. From the viewpoint of the equilibrium type, they are classified into systems with no equilibria, with a line of equilibrium points, and with one stable equilibrium. The type of nonlinearity presents another method of categorization. System properties are explored versus the different parameters to identify the values corresponding to the presence of strange attractors. The behavior of the systems is explored for integer order and fractional order derivatives using the suitable numerical techniques. The studied properties include time series, phase portraits, and maximum Lyapunov exponent. © 2018 Elsevier Inc. All rights reserved.
Chaos and bifurcation in controllable jerk-based self-excited attractors
In the recent decades, utilization of chaotic systems has flourished in various engineering applications. Hence, there is an increasing demand on generalized, modified and novel chaotic systems. This chapter combines the general equation of jerk-based chaotic systems with simple scaled discrete chaotic maps. Two continuous chaotic systems based on jerk-equation and discrete maps with scaling parameters are presented. The first system employs the scaled tent map, while the other employs the scaled logistic map. The effects of different parameters on the type of the response of each system are investigated through numerical simulations of time series, phase portraits, bifurcations and Maximum Lyapunov Exponent (MLE) values against all system parameters. Numerical simulations show interesting behaviors and dependencies among these parameters. Analogy between the effects of the scaling parameters is presented for simple one-dimensional discrete chaotic systems and the continuous jerk-based chaotic systems with more complicated dynamics. The impacts of these scaling parameters appear on the effective ranges of other main system parameters and the ranges of the obtained solution. The dependence of equilibrium points on the sign of one of the scaling parameters results in coexisting attractors according to the signs of the parameter and the initial point. In addition, switching can be used to generate double-scroll attractors. Moreover, bifurcation and chaos are studied for fractional-order of the derivative. © 2018, Springer International Publishing AG.
Applications of continuous-time fractional order chaotic systems
The study of nonlinear systems and chaos is of great importance to science and engineering mainly because real systems are inherently nonlinear and linearization is only valid near the operating point. The interest in chaos was increased when Lorenz accidentally discovered the sensitivity to initial condition during his simulation work on weather prediction. When a nonlinear system is exhibiting deterministic chaos, it is very difficult to predict its response under external disturbances. This behavior is a double-edged weapon. From a control and synchronization point of view, this proposes a challenge. On the other hand, from a communications and encryption perspective, this provides a higher level of security. This chapter is a survey of the recent contributions in engineering applications of fractional order chaotic continuous-time systems. The applications include but not limited to: communication and encryption, FPGA implementations, synchronization and control, modeling of electric motors, and biomedical applications. © 2018 Elsevier Inc. 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.