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Abstract This paper discusses the synchronization problem of N-coupled fractional-order chaotic systems with ring connection via bidirectional coupling. On the basis of the direct design method, we design the appropriate controllers to transform the fractional-order error dynamical system into a nonlinear system with antisymmetric structure. By choosing appropriate fractional-order Lyapunov functions and employing the fractional-order Lyapunov-based stability theory, several sufficient conditions are obtained to ensure the asymptotical stabilization of the fractional-order error system at the origin. The proposed method is universal, simple, and theoretically rigorous. Finally, some numerical examples are presented to illustrate the validity of theoretical results.

This research aims to discuss and control the chaotic behaviour of an autonomous fractional biological oscillator. Indeed, the concept of fractional calculus is used to include memory in the modelling formulation. In addition, we take into account a new auxiliary parameter in order to keep away from dimensional mismatching. Further, we explore the chaotic attractors of the considered model through its corresponding phase-portraits. Additionally, the stability and equilibrium point of the system are studied and investigated. Next, we design a feedback control scheme for the purpose of chaos control and stabilization. Afterwards, we introduce an efficient active control method to achieve synchronization between two chaotic fractional biological oscillators. The efficiency of the proposed stabilizing and synchronizing controllers is verified via theoretical analysis as well as simulations and numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an adaptive neural network (NN) synchronization controller is designed for two identical strict-feedback chaotic systems (SFCSs) subject to dead-zone input. The dead-zone models together with the system uncertainties are approximated by NNs. The dynamic surface control (DSC) approach is applied in the synchronization controller design, and the traditional problem of “explosion of complexity” that usually occurs in the backstepping design can be avoided. The proposed synchronization method guarantees the synchronization errors tend to an arbitrarily small region. Finally, this paper presents two simulation examples to confirm the effectiveness and the robustness of the proposed control method.

Abstract Observer design for nonlinear systems is very important in state-based stabilization, fault detection, chaos synchronization and secret communication. This paper deals with synchronization problem of a class of fractional-order neural networks (FONNs) based on system observer. Two sufficient conditions are given for the FONNs with known constant parameters and unknown time-varying parameters, respectively. Based on the fractional Lyapunov stability criterion, the proposed sliding mode observer can guarantee that the synchronization error between two identical FONNs converges to zero asymptotically, and all involved signals keep bounded. Finally, some simulation examples are provided to indicate the effectiveness of the proposed method.

Abstract The topic related to the coexistence of different synchronization types between fractional-order chaotic systems is almost unexplored in the literature. Referring to commensurate and incommensurate fractional systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between nonidentical systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. The approach, which can be applied to a wide class of chaotic/hyperchaotic fractional-order systems in the master-slave configuration, is based on two new theorems involving the fractional Lyapunov method and stability theory of linear fractional systems. Two examples are provided to highlight the capability of the conceived method. In particular, referring to commensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Rössler system of order 2.7 and the hyperchaotic four-dimensional Chen system of order 3.84. Finally, referring to incommensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Lü system of order 2.955 and the hyperchaotic four-dimensional Lorenz system of order 3.86.

This paper proposes a modified adaptive sliding-mode control technique and investigates the reduced-order and increased-order synchronization between two different fractional-order chaotic systems using the master and slave system synchronization arrangement. The parameters of the master and slave systems are different and uncertain. These systems exhibit different chaotic behavior and topological properties. The dynamic behavior of the proposed synchronization schemes is more complex and unpredictable. These attributes of the proposed synchronization schemes enhance the security of the information signal in digital communication systems. The proposed switching law ensures the convergence of the error vectors to the switching surface and the feedback control signals guarantee the fast convergence of the error vectors to the origin. Lyapunov stability theory proves the asymptotic stability of the closed-loop. The paper also designs suitable parameters update laws the estimate the unknown parameters. Computer-based simulation results verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate hybrid adaptive synchronization issue for a class of perturbed fractional-order chaotic systems with nondeterministic nonlinear terms. On the basis of fractional-order extended version of Lyapunov stability criterion, a novel fuzzy adaptive synchronization control protocol coupled with backstepping-based method is constructed, ensuring that the synchronization errors converge to a sufficiently small region of the origin. In order to avert the occurrence of "explosion of complexity", we take advantage of a fuzzy logic system to estimate the unknown systematic term approximately in every backstepping step. Finally, some numerical simulations are given to exemplify the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper presents new results related to the coexistence of function-based hybrid synchronization types between non-identical incommensurate fractional-order systems characterized by different dimensions and orders. Specifically, a new theorem is illustrated, which ensures the coexistence of the full-state hybrid function projective synchronization (FSHFPS) and the inverse full-state hybrid function projective synchronization (IFSHFPS) between wide classes of three-dimensional master systems and four-dimensional slave systems. In order to show the capability of the approach, a numerical example is reported, which illustrates the coexistence of FSHFPS and IFSHFPS between the incommensurate chaotic fractional-order unified system and the incommensurate hyperchaotic fractional-order Lorenz system. [ABSTRACT FROM AUTHOR]

The topic related to the coexistence of different synchronization types between fractional-order chaotic systems is almost unexplored in the literature. Referring to commensurate and incommensurate fractional systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between nonidentical systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization ( IS), antiphase synchronization ( AS), and inverse full state hybrid projective synchronization ( IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. The approach, which can be applied to a wide class of chaotic/hyperchaotic fractional-order systems in the master-slave configuration, is based on two new theorems involving the fractional Lyapunov method and stability theory of linear fractional systems. Two examples are provided to highlight the capability of the conceived method. In particular, referring to commensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Rössler system of order 2.7 and the hyperchaotic four-dimensional Chen system of order 3.84. Finally, referring to incommensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Lü system of order 2.955 and the hyperchaotic four-dimensional Lorenz system of order 3.86. [ABSTRACT FROM AUTHOR]

There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes. [ABSTRACT FROM AUTHOR]

In this paper, a memristor-based hyperchaotic system is introduced. Considering time delays between the drive system and the response system in the process of synchronization, this paper designs one kind of flexible sandwich controller, which includes a rest in the sandwich structure, to realize the synchronization between two memristor-based hyperchaotic systems. Based on Lyapunov stability theory, matrix inequality, sandwich control and considering time delays, the exponential synchronization conditions for the memristor-based hyperchaotic systems with time delays via sandwich control are given. Finally, simulation results are displayed to verify the effectiveness and feasibility of this method. [ABSTRACT FROM AUTHOR]

The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out, in this paper, we study the bounds of a high-order Lorenz-Stenflo system arising in mathematical physics. Based on Lyapunov stability theory, we show that there exists a globally exponential attractive set for this system. The innovation of the paper is that we not only prove that this system is globally bounded for all the parameters, but also give a family of mathematical expressions of global exponential attractive sets of this system with respect to its parameters. We also study some other dynamical characteristics of this chaotic system such as invariant sets and chaotic behaviors. To justify the theoretical analysis, we carry out detailed numerical simulations. [ABSTRACT FROM AUTHOR]

This paper investigates the synchronization of multiple chaotic systems with unknown parameters using adaptive control method, and two kinds of different synchronization modes are considered here. One is that more response systems synchronize with one drive system, and the other is the ring transmission synchronization, which guarantees that all chaotic systems can synchronize with each other. The definition of adaptive synchronization of multiple chaotic systems with unknown parameters is given, and then based on the idea of adaptive control method, adaptive laws are derived to estimate the unknown parameters, and nonlinear adaptive controllers are developed to ensure the asymptotical stability of two classes of error systems. Finally, simulation results are presented to verify the effectiveness of proposed synchronization schemes. [ABSTRACT FROM AUTHOR]

Discrete epidemic models are popularly used to detect the pathogenesis, spreading, and controlling of the diseases. The three-dimensional discrete ${SIRS}$ epidemic models are more suitable than the two-dimensional discrete models to describe the spreading characters of the diseases. In this paper, the complex dynamical behaviors of a three-dimensional discrete ${SIRS}$ epidemic model with standard incidence rate are discussed. We choose the time step size parameter as a bifurcation parameter, the existence, stability, and direction of Hopf bifurcation are proved by using the normal form theorem and bifurcation theory. Moreover, the numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors, such as the invariant cycle, period-7 orbits and period-12 orbits with more than one attractors and chaotic sets. The flip bifurcation caused by the step size parameter is also obtained by a numerical simulation. Most importantly, when the adequate contact rate and the death rate of the infective individuals are chosen as the bifurcation parameters, there also exist a Hopf bifurcation, a flip bifurcation, chaos, and strange attractors. These results provide significant information for the disease controlling when there appear complex dynamical behaviors in the epidemic model. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order model of a financial risk dynamical system is proposed and the complex behavior of such a system is presented. The basic dynamical behavior of this financial risk dynamic system, such as chaotic attractor, Lyapunov exponents, and bifurcation analysis, is investigated. We find that numerical results display periodic behavior and chaotic behavior of the system. The results of theoretical models and numerical simulation are helpful for better understanding of other similar nonlinear financial risk dynamic systems. Furthermore, the adaptive fuzzy control for the fractional-order financial risk chaotic system is investigated on the fractional Lyapunov stability criterion. Finally, numerical simulation is given to confirm the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the problem of synchronization for chaotic neural networks with stochastic disturbances. Firstly, we provide a basic result that the systems including the drive system, response system, and error system have a unique solution on the whole time horizon. Based on this result, we design a new control law such that the response system can be synchronized with the drive chaotic system in finite time. Furthermore, we show that the settling time is independent of the initial data under some proper conditions, which hints that the fixed-time synchronization of chaotic neural networks can be realized by our proposed method. Finally, we give simulations to verify the theoretical analysis for our main results. [ABSTRACT FROM AUTHOR]

In this study, we examine the stabilization of fractional-order chaotic nonlinear dynamical systems with model uncertainties and external disturbances. We used the sliding mode controller by a new approach for controlling and stabilization of these systems. In this research, we replaced a continuous function with the sign function in the controller design and the sliding surface to suppress chattering and undesirable vibration effects. The advantages of the proposed control method are rapid convergence to the equilibrium point, the absence of chattering and unwanted oscillations, high resistance to uncertainties, and the possibility of applying this method to most fractional order chaotic systems. We applied the direct method of Lyapunov stability theory and the frequency distributed model to prove the stability of the slip surface and closed loop system. Finally, we simulated this method on two commonly used and practical chaotic systems and presented the results. [ABSTRACT FROM AUTHOR]

This paper introduces the projective synchronization of different fractional-order multiple chaotic systems with uncertainties, disturbances, unknown parameters, and input nonlinearities. A fractional adaptive sliding surface is suggested to guarantee that more slave systems synchronize with one master system. First, an adaptive sliding mode controller is proposed for the synchronization of fractional-order multiple chaotic systems with unknown parameters and disturbances. Then, the synchronization of fractional-order multiple chaotic systems in the presence of uncertainties and input nonlinearity is obtained. The developed method can be used for many of fractional-order multiple chaotic systems. The bounds of the uncertainties and disturbances are unknown. Suitable adaptive rules are established to overcome the unknown parameters. Based on the fractional Lyapunov theorem, the stability of the suggested technique is proved. Finally, the simulation results demonstrate the feasibility and robustness of our suggested scheme. [ABSTRACT FROM AUTHOR]