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Recently, a new hyperchaos generator, obtained by controlling a three-dimensional autonomous chaotic system — Chen's system — with a periodic driving signal, has been found. In this letter, we formulate and study the hyperchaotic behaviors in the corresponding fractional-order hyperchaotic Chen's system. Through numerical simulations, we found that hyperchaos exists in the fractional-order hyperchaotic Chen's system with order less than 4. The lowest order we found to have hyperchaos in this system is 3.4. Finally, we study the synchronization problem of two fractional-order hyperchaotic Chen's systems.

In this paper, a novel digital secure communication scheme is firstly proposed. Different from the usual secure communication schemes based on chaotic synchronization, the proposed scheme employs asynchronous communication which avoids the weakness of synchronous systems and is susceptible to environmental interference. Moreover, as to the transmission errors and data loss in the process of communication, the proposed scheme has the ability to be error-checking and error-correcting in real time. In order to guarantee security, the fractional-order complex chaotic system with the shifting of order is utilized to modulate the transmitted signal, which has high nonlinearity and complexity in both frequency and time domains. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the scheme.

In this paper, aiming at the problems of slow estimation speed and low estimation precision of traditional fractional-order system (FOS) parameter estimation method, an improved Archimedes optimization algorithm (IAOA) is proposed to calculate the optimal value. By establishing the parameter estimation model and the cost function, the parameter estimation problem is formulated as an optimization problem. As opposed to the Archimedes optimization algorithm (AOA), the IAOA introduces three improvements: leadership behavior, levy flight behavior and a new adaptive strategy. This paper verifies the performance of the IAOA by selecting 10 classic test functions. IAOA is applied to the parameter estimation problem of fractional-order unified system to verify the accuracy and feasibility of the algorithm. The simulation results prove that the IAOA has better global optimization ability and estimation accuracy than the original algorithm.

In this paper, we numerically investigate the chaotic behaviors of the fractional-order Chua's circuit with a piecewise-linear nonlinearity. We find that chaos exists in the fractional-order Chua's circuit with order less than 3. The lowest order we find to have chaos is 2.7 in the homogeneous fractional-order Chua's circuit and 2.8 in the unhomogeneous fractional-order Chua's circuit. Our results are validated by the existence of a positive Lyapunov exponent. A chaos synchronization method is also presented for synchronizing the homogeneous fractional-order chaotic Chua's systems. The approach, based on stability theory of fractional-order linear systems, is simple and theoretically rigorous. It does not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization method.

In this paper, in order to analyze the coexistent multiple-stability of system, a fractional-order memristive Chua’s circuit with time delay is proposed, which is composed of a passive flux-controlled memristor and a negative conductance as a parallel combination. First, the Chua’s circuit can be considered as a nonlinear feedback system consisting of a nonlinear block and a linear block with low-pass properties. In the complex plane, the nonlinear element of the system can be approximated by a variable gain called a describing function. Second, compared with conventional computation, the describing function can accurately predict the hidden dynamics, fixed points, periodic orbits, unstable behaviors of the system. By using this method, the full mapping of the system dynamics in parameter spaces is presented, and the coexistent multiple-stability of the system is investigated in detail. Third, using bifurcation diagram, phase diagram, time domain diagram and power spectrum diagram, the dynamical behaviors of the system under different system parameters and initial values are discussed. Finally, based on Adams–Bashforth–Moulton (ABM) method, the correctness of theoretical analysis is verified by numerical simulation, which shows that the fractional-order delayed memristive Chua’s system has complex coexistent multiple-stability.

This paper presents dynamic behavior of a fractional-order memristive time-delay system and its application in image encryption. First, a fractional-order memristive time-delay system is proposed, and the stability and bifurcation behaviors of the system are theoretically analyzed. Some limited conditions for describing the stability interval and switching between different dynamic behaviors are derived. Second, the dynamic characteristics of the system are analyzed through the coexisting attractors, coexisting bifurcation diagrams, the Largest Lyapunov exponents (LLE), the 0-1 test. When parameters change, such as time delay and fractional order, the system transits from steady state to periodic state, single scroll chaotic state, double scroll chaotic state. Furthermore, an image encryption scheme based on the fractional-order memristive time-delay system is introduced, and some statistical features are analyzed. Finally, numerical simulations verify the validity of the theoretical analysis and safety of the image encryption scheme based on the fractional-order delayed memristive chaotic system.

Realization of fractional-order double-scroll chaotic system using Operational Transconductance Amplifiers (OTAs) as active elements are presented in this paper. The fractional-order double-scroll chaotic system has been studied before as well using passive RC-ladder and tree-based structures but in this paper the requisite fractional-order integration has been accomplished through an integer-order multiple-feedback topology. As compared to double or multiple scroll chaotic systems existing in the open literature, the proposed realization offers the advantages of (a) low-voltage implementation, (b) integrablity as the design is resistor- and inductor-less and only grounded components have been employed in the design, and, (c) electronic tunability of the fractional order, time-constants and gain factors. In order to demonstrate the usefulness of the chaotic system, a simple secure message communication system has been designed and verified for its operation. The theoretical predictions of the proposed implementations have been verified by using 0.35$\mu$m complementary metal oxide semiconductor (CMOS) process file provided by Austrian Micro System (AMS).

The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived analytically to achieve synchronization of the chaotic fractional-order Chen, Rössler and modified Chua systems. Numerical simulations are provided to verify the theoretical analysis.

This paper presents a fractional-order population model which consists of the two-predators and one-prey scheme. For this new model, the numerical solution is derived and the simulations are performed for various sets of model parameters together with stability analysis for commensurate and incomensurate orders of the fractional-order population model. The results obtained via the simulations show that chaos can be observed in such population model.

In this paper, we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piecewise continuous functions. By using techniques from the theory of differential inclusions, the underlying piecewise functions can be locally or globally approximated. The approximation results can be used to model piecewise continuous-time dynamical systems of integer or fractional-order. In this way, by overcoming the lack of numerical methods for differential equations of fractional-order with discontinuous right-hand side, unattainable procedures for systems modeled by this kind of equations, such as chaos control, synchronization, anticontrol and many others, can be easily implemented. Several examples are presented and three comparative applications are studied.

In this paper, the effect of the parameter switching (PS) algorithm in a fractional order chaotic circuit is investigated both in simulation and experiment. The Chen system of fractional order is focused and realized in an electronic circuit. By designing a switching circuit, the PS algorithm is implemented and it is the first time, the paradoxical “Chaos $+$ Chaos $=$ Order” is presented in an electronic circuit. Both the simulation and experimental results confirm that the obtained attractor under switching approximates the attractor of the time-averaged model. Some important design issues for the circuitry realization of the PS scheme are pointed out. Finally, our work confirms the practical usage of PS algorithm in potential applications such as attractor synthesis and chaos control.

In this paper, we investigate chaos and adaptive control of the magnetic-field electromechanical transducer wherein the electric characteristics of the capacitor contain the fractional-order derivative. The phase diagrams for different values of the fractional-order exhibit chaotic characteristics of the magnetic-field electromechanical transducer. In the process of controller design, a continuous frequency distributed model is utilized to construct the indirect Lyapunov stability criterion and a Chebyshev neural network with weight, and fractional-order adaptive law is introduced to approximate the complicated unknown function. To suppress chaotic oscillation, an adaptive control scheme by fusing Chebyshev neural network and backstepping is presented to guarantee that the closed-loop system is globally asymptotically stable. To illustrate the feasibility of the proposed approach, simulation studies are done in the end.

A novel three-dimensional fractional-order autonomous chaotic system marked by the ample and complex coexisting attractors is presented. There are a total of seven terms including four nonlinearities in the new system. The evolution of coexisting attractors of the system are numerically investigated by considering both the fractional-order and other system parameters as bifurcation parameters. Numerical simulation results indicate that the system has a huge amount of multifarious coexisting strange attractors for various ranges of parameters, including coexisting point, periodic attractors, multifarious coexisting chaotic, and periodic attractors. Compared with other chaotic systems, the biggest difference and most attractive feature is the capability of the proposed fractional-order system to produce coexisting attractors that undergo a simultaneous displacement phenomenon with variation of a single parameter. Moreover, it is worth noting that constant Lyapunov exponents and the interesting phenomenon of transient coexisting attractors are also observed. Finally, the corresponding implementation circuit is designed. The consistency of the hardware experimental results with numerical simulations verifies the feasibility of the new fractional-order chaotic system.

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an $S$-asymptotically $T$-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

A stochastic vibro-impact system has triggered a consistent body of research work aimed at understanding its complex dynamics involving noise and nonsmoothness. Among these works, most focus is on integer-order systems with Gaussian white noise. There is no report yet on response analysis for fractional-order vibro-impact systems subject to colored noise, which is presented in this paper. The biggest challenge for analyzing such systems is how to deal with the fractional derivative of absolute value functions after applying nonsmooth transformation. This problem is solved by introducing the Fourier transformation and deriving the approximate probabilistic solution of the fractional-order vibro-impact oscillator subject to colored noise. The reliability of the developed technique is assessed by numerical solutions. Based on the theoretical result, we also present the critical conditions of stochastic bifurcation induced by system parameters and show bifurcation diagrams in two-parameter planes. In addition, we provide a stochastic bifurcation with respect to joint probability density functions. We find that fractional order, coefficient of restitution factor and correlation time of colored noise excitation can induce stochastic bifurcations.

In this paper, the Benettin–Wolf algorithm for determining all Lyapunov exponents of noncommensurate fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. The paper continues the work started in [Danca & Kuznetsov, 2018], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended systems, the Adams–Bashforth–Moulton scheme for fractional differential equations is utilized. Like the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as a function of orders, or a function of a bifurcation parameter. Special attention is paid to the periodicity of fractional-order systems and its influences. The case of noncommensurate Lorenz system is demonstrated.

This paper delves into the topics of Finite-Time Stabilization (FTS) and Finite-Time Contractive Stabilization (FTCS) for Fractional-Order Nonlinear Systems (FONSs). To address these issues, we employ a State-Dependent Delayed Impulsive Controller (SDDIC). By leveraging both Lyapunov theory and impulsive control theory, we establish sufficient conditions for achieving stability criteria for fractional-order systems. Initially, we employ the aforementioned sufficient conditions to derive stability criteria for general FONSs within the SDDIC framework, employing Linear Matrix Inequality (LMI) techniques. Furthermore, we apply these theoretical findings to tackle the challenge of finite-time synchronization in fractional-order chaotic systems using the proposed SDDIC. We substantiate the efficacy of these theoretical advancements through numerical simulations that vividly demonstrate their capability to achieve finite-time synchronization in fractional-order cellular neural networks and fractional-order Chua’s circuits. Moreover, we introduce an innovative chaos-based multi-image encryption algorithm, thereby contributing significantly to the field. To ensure the algorithm’s robustness, we subject it to rigorous statistical tests, which confidently affirm its capacity to provide the requisite level of security.

This paper excogitates a bifurcation control strategy for a delayed fractional-order population dynamics model with incommensurate orders. First and foremost, by using stability theory of fractional differential equations, the sufficient conditions for the stability of the positive equilibrium are established. It is not difficult to find that the fractional-order system has a wider stability region than the traditional integer-order system. Second, taking time delay as bifurcation parameter, the sufficient criteria for Hopf bifurcation are obtained. In the next place, it is interesting to introduce a delayed feedback controller to guide Hopf bifurcation. The results reveal that the bifurcation dynamics of the model could be effectively controlled as long as the delay or fractional order is carefully adjusted. In conclusion, numerical simulations are carried out to confirm our theoretical results.

In this study, a synchronization problem for spatio-temporal partial differential systems is addressed and researched within a subjectivist framework. In light of Lyapunov direct method and some proposed nonlinear controllers, a new scheme is established to accomplish a full synchronization between two reaction–diffusion systems of integer- and fractional-order. In particular, a novel vector-valued control law is analytically derived to attain the desired synchronization between two chemical models, namely, the Lengyel–Epstein and Gray–Scott models. To validate the obtained theoretical results, further numerical simulations are carried out in 2D and 3D configurations.

Our study focuses on the analysis of asymptotic stability and the development of asymptotic stabilization control methods for fractional vehicle suspension systems (FVSS). We begin by constructing a mathematical model for FVSS using the state-space equations of Caputo fractional calculus. Initially, we utilize the fractional Routh–Hurwitz criterion to derive the necessary conditions for asymptotic stability and instability in the open-loop system of the fractional vehicle suspension. Subsequently, we propose a novel control strategy for FVSS and establish an associated asymptotic stabilization criterion by combining a new vector Lyapunov function with the $M$-matrix method. Moreover, we extend the fractional-order vehicle suspension model to include time delay resulting from the interactions between different variables in the real system, thus creating a FVSS with time delay. Based on the vector Lyapunov function, $M$-matrix measure, and Razumikhin interpretation, we develop a control strategy specifically tailored for FVSSs with time delay. Lastly, we compare two numerical simulations of the FVSS, one with time delay and one without, to demonstrate the accuracy, effectiveness, and applicability of the proposed method presented in our paper.