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.
Carbon Nanomaterials and Their Composites as Adsorbents
Carbon nanomaterials with various nanostructures (carbon nanotubes, graphene, graphene oxide, fullerene, nano diamonds, carbon quantum dots, carbon nanofibers, graphitic carbon nitrides, and nano porous carbons) are the decade’s most distinguishing and popular materials. They have distinctive physicochemical qualities such as chemical stability, mechanical strength, hardness, thermal and electrical conductivities, and so on. Furthermore, they are easily surface functionalized and tweaked, modifying them for high-end specific applications. Carbon nanostructures’ properties and surface characteristics are determined by the synthesis method used to create them. Nanoscience and nanotechnology have the potential to create materials with unexpected functions and qualities, which are transforming all industries. Carbon nanoparticles such as fullerene, carbon nanotubes, and graphene stand out among the various kinds of nanomaterials. These nanoparticles offer a wide range of practical applications, particularly in adsorption processes. Carbon nanoparticles exhibit unique structures that could be used in the construction of extremely sensitive, selective, and effective adsorbent devices for the removal of inorganic, organic, and biological pollutants from water solutions, as well as nano sensors and drug delivery systems. In this chapter, we demonstrated the number of studies published in recent years that used carbon nanomaterials as adsorbents. Furthermore, this chapter discusses essential features of adsorption and different nanocarbon carbon composite material, such as the contrast between physical and chemical absorption. Furthermore, diverse carbon nanomaterial synthesis such as AC–FeO ?Cu and Bimetallic FeO ?Cu/algae activated carbon composites AC–Fe0 ?Cu methodologies, functionalization, and characteristics are provided and logically addressed. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
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.
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.
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.