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.
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.
Memristor and inverse memristor: Modeling, implementation and experiments
Pinched hysteresis is considered to be a signature of the existence of memristive behavior. However, this is not completely accurate. In this chapter, we are discussing a general equation taking into consideration all possible cases to model all known elements including memristor. Based on this equation, it is found that an opposite behavior to the memristor can exist in a nonlinear inductor or a nonlinear capacitor (both with quadratic nonlinearity) or a derivative-controlled nonlinear resistor/transconductor which we refer to as the inverse memristor. We discuss the behavior of this new element and introduce an emulation circuit to mimic its behavior. Connecting the conventional elements with the memristor and/or with inverse memeristor either in series or parallel affects the pinched hysteresis lobes where the pinch point moves from the origin and lobes’ area shrinks or widens. Different cases of connecting different elements are discussed clearly especially connecting the memristor and the inverse memristor together either in series or in parallel. New observations and conditions on the memristive behavior are introduced and discussed in detail with different illustrative examples based on numerical, and circuit simulations. © Springer International Publishing AG 2017.
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.
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.
A study on coexistence of different types of synchronization between different dimensional fractional chaotic systems
In this study, robust approaches are proposed to investigate the problem of the coexistence of various types of synchronization between different dimensional fractional chaotic systems. Based on stability theory of linear fractional order systems, the co-existence of full state hybrid function projective synchronization (FSHFPS), inverse generalized synchronization (IGS), inverse full state hybrid projective synchronization (IFSHPS) and generalized synchronization (GS) is demonstrated. Using integer-order Lyapunov stability theory and fractional Lyapunov method, the co-existence of FSHFPS, inverse full state hybrid function projective synchronization (IFSHFPS), IGS and GS is also proved. Finally, numerical results are reported, with the aim to illustrate the capabilities of the novel schemes proposed herein. © Springer International Publishing AG 2017. 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.
Preparation and Characterization of nZVI, Bimetallic Fe 0-Cu, and Fava Bean Activated Carbon-Supported Bimetallic AC-F e 0-Cu for Anionic Methyl Orange Dye Removal
Nano zero-valent iron (nZVI), bimetallic Nano zero-valent iron-copper (Fe 0- Cu), and fava bean activated carbon-supported with bimetallic Nano zero-valent iron-copper (AC-F e 0-Cu) were prepared and characterized by DLS, FT-IR, XRD, and SEM. The influence of the synthesized adsorbents on the adsorption and removal of soluble anionic methyl orange (M.O) dye was investigated using UV-V spectroscopy. The influence of numerous operational parameters was studied at varied pH (3–9), time intervals (15–180 min), and dye concentrations (25–1000 ppm) to establish the best removal conditions. The maximum removal efficiency of M.O. using the prepared adsorbent materials reached about 99%. The removal efficiency is modeled using response surface methodology (RSM). The Bimetallic Fe -Cu, the best experimental and predicted removal efficiency is 96.8% RE. For the H2SO4 chemical AC- Fe -Cu, the best experimental and removal efficiency is 96.25% RE. The commercial AC-Fe0–Cu, the best experimental and predicted removal efficiency is 94.93%RE. This study aims to produce low-cost adsorbents such as Bimetallic Fe0-Cu, and Fava Bean Activated Carbon-Supported Bimetallic AC-Fe0-Cu to treat the industrial wastewater from the anionic methyl orange (M.O) dye and illustrate its ability to compete H2SO4 chemical AC-Fe0-Cu, and commercial AC-Fe0-Cu. © 2023, The Author(s).
Crystal violet removal using bimetallic Fe0–Cu and its composites with fava bean activated carbon
Nano zero-valent iron (nZVI), bimetallic nano zero-valent iron-copper (Fe0– Cu), and fava bean activated carbon-supported bimetallic nano zero-valent iron-copper (AC-Fe0-Cu) are synthesized and characterized using DLS, zeta potential, FT-IR, XRD, and SEM. The maximum removal capacity is demonstrated by bimetallic Fe0–Cu, which is estimated at 413.98 mg/g capacity at pH 7, 180 min of contact duration, 120 rpm shaking speed, ambient temperature, 100 ppm of C.V. dye solution, and 1 g/l dosage. The elimination capability of the H2SO4 chemical AC-Fe0-Cu adsorbent is 415.32 mg/g under the same conditions but with a 150 ppm C.V. dye solution. The H3PO4 chemical AC-Fe0-Cu adsorbent achieves a removal capacity of 413.98 mg/g under the same conditions with a 350 ppm C.V. dye solution and a 1.5 g/l dosage. Optimal conditions for maximum removal efficiency are determined by varying pH (3–9), time intervals (15–180 min), and initial dye concentrations (25–1000 ppm). Kinetic and isothermal models are used to fit the results of time and concentration experiments. The intra-particle model yields the best fit for bimetallic Fe0–Cu, H2SO4 chemical AC- Fe0–Cu, and H3PO4 chemical AC-Fe0-Cu, with corrected R-Squared values of 0.9656, 0.9926, and 0.964, respectively. The isothermal results emphasize the significance of physisorption and chemisorption in concentration outcomes. Response surface methodology (RSM) and artificial neural networks (ANN) are employed to optimize the removal efficiency. RSM models the efficiency and facilitates numerical optimization, while the ANN model is optimized using the moth search algorithm (MSA) for optimal results. © 2023
A review of coagulation explaining its definition, mechanism, coagulant types, and optimization models; RSM, and ANN
The textile business is one of the most hazardous industries since it produces several chemicals, such as dyes, which are released into water streams with ef-fluents. For the survival of the planet’s life and the advancement of humanity, water is a crucial resource. One of the anthropogenic activities that pollute and consume water is the textile industry. Thus, the purpose of the current effort is to Apply coagulation as a Physico-chemical and biological treatment strat-egy with different techniques and mechanisms to treat the effluent streams of textile industries. The discharge of these effluents has a negative impact on the environment, marine life, and human health. Therefore, the treatment of these effluents before discharging is an important matter to reduce their adverse ef-fect. Many physico-chemical and biological treatment strategies for contaminants removal from polluted wastewater have been proposed. Coagulation is thought to be one of the most promising physico-chemical strategies for removing con-taminants and colouring pollutants from contaminated water. Coagulation is accompanied by a floculation process to aid precipitation, as well as the collection of the created sludge following the treatment phase. Different commercial, and natural coagulants have been applied as a coagulants in the process of coagulation. Additionally, many factors such as; pH, coagulant dose, pollu-tants concentration are optimized to obtain high coagulants removal capacity. This review will discuss the coagulation process, coagulant types and aids in addition to the factors affecting the coagulation process. Additionally, a brief comparison between the coagulation process, and the other processes; princi-ple, advantages, disadvantages, and their efficiency were discussed throgh the review. Furthermore, it discusses the models and optimization techniques used for the coagulation process including response surface methodology (RSM), ar-tificial neural network (ANN), and several metaheuristic algorithms combined with ANN and RSM for optimization in previous work. The ANN model has more accurate results than RSM. The ANN combined with genetic algorithm gives an accurate predicted optimum solution. © 2023 The Authors