Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lü chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen??s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen??s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

Generalized projective synchronization of the fractional-order Chen hyperchaotic system

In this paper, we numerically investigate the hyperchaotic behaviors in the fractional-order Chen hyperchaotic systems. By utilizing the fractional calculus techniques, we find that hyperchaos exists in the fractional-order Chen hyperchaotic system with the order less than 4. We found that the lowest order for hyperchaos to have in this system is 3.72. Our results are validated by the existence of two positive Lyapunov exponents. The generalized projective synchronization method is also presented for synchronizing the fractional-order Chen hyperchaotic systems. The present technique is based on the Laplace transform theory. This simple and theoretically rigorous synchronization approach enables synchronization of fractional-order hyperchaotic systems to be achieved and does not require the computation of the conditional Lyapunov exponents. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives

This paper addresses the problem of synchronization of chaotic fractional-order systems with different orders of fractional derivatives. Based on the stability theory of fractional-order linear systems and the idea of tracking control, suitable controllers are correspondingly proposed for two cases: the first is synchronization between two identical chaotic fractional-order systems with different fractional orders, and the other is synchronization between two nonidentical fractional-order chaotic systems with different fractional orders. Three numerical examples illustrate that fast synchronization can be achieved even between a chaotic fractional-order system and a hyperchaotic fractional-order system.     

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.     

A practical synchronization approach for fractional-order chaotic systems

A practical synchronization approach is proposed for a class of fractional-order chaotic systems to realize perfect \(\delta \)-synchronization, and the nonlinear functions in the fractional-order chaotic systems are all polynomials. The \(\delta \)-synchronization scheme in this paper means that the origin in synchronization error system is stable. The reliability of \(\delta \)-synchronization has been confirmed on a class of fractional-order chaotic systems with detailed theoretical proof and discussion. Furthermore, the \(\delta \)-synchronization scheme for the fractional-order Lorenz chaotic system and the fractional-order Chua circuit is presented to demonstrate the effectiveness of the proposed method.     

Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations

Chaotic systems in practice are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems. Based on Lyapunov stability theory and a fractional-order differential inequality, a modified adaptive control scheme and adaptive laws of parameters are developed to robustly synchronize coupled fractional-order chaotic systems with unknown parameters and uncertain perturbations. This synchronization approach is simple, global and theoretically rigorous. Simulation results for two fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Chaos detection and parameter identification in fractional-order chaotic systems with delay

The paper first applies the 0–1 test for chaos to detecting chaos exhibited by fractional-order delayed systems. The results of the test reveal that there exists chaos in some fractional-order delayed systems with specific parameter values, which coincides with previous reports based on the phase portrait. In addition, it is very important to identify exactly the unknown specific parameters of fractional-order chaotic delayed systems in chaos control and synchronization. Thus, a method for parameter identification of fractional-order chaotic delayed systems based on particle swarm optimization (PSO) is presented. By treating the orders as parameters, the parameters and orders are identified through minimizing an objective function. PSO can efficiently find the optimal feasible solution of the objective function. Finally, numerical simulations on fractional-order chaotic logistic delayed system and fractional-order chaotic Chen delayed system show that the proposed method has effective performance of parameter identification.     

Stability analysis for nonlinear fractional-order systems based on comparison principle

This work constructs a theoretical framework for the stability analysis of nonlinear fractional-order systems. A new definition, the generalized Caputo fractional derivative, is proposed for the first time. Based on that, the comparison principles for scalar and vector fractional-order systems are constructed, respectively. Furthermore, a sufficient theorem for stability analysis is proved, and how to use this theorem in stabilization is also discussed. Three examples have been presented to illustrate how to use the developed theory to analyze the stability and to design stabilization controllers. With the proposed method, the problems of stabilization and synchronization of the fractional-order chaotic fractional-order systems can be easily solved with linear feedback control.     

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lü system, and fractional-order Duffing system.     

Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems

Recently, the fractional-order Chen–Lee system was proven to exhibit chaos by the presence of a positive Lyapunov exponent. However, the existence of chaos in fractional-order Chen–Lee systems has never been theoretically proven in the literature. Moreover, synchronization of chaotic fractional-order systems was extensively studied through numerical simulations in some of the literature, but a theoretical analysis is still lacking. Therefore, we devoted ourselves to investigating the theoretical basis of chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems in this paper. Based on the stability theorems of fractional-order systems, the necessary conditions for the existence of chaos and the controllers for hybrid projective synchronization were derived. The numerical simulations show coincidence with the theoretical results.     

Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems

This article examines the synchronization performance between two fractional-order systems, viz., the Ravinovich?CFabrikant chaotic system as drive system and the Lotka?CVolterra system as response system. The chaotic attractors of the systems are found for fractional-order time derivatives described in Caputo sense. Numerical simulation results which are carried out using Adams?CBoshforth?CMoulton method show that the method is reliable and effective for synchronization of nonlinear dynamical evolutionary systems. Effects on synchronization time due to the presence of fractional-order derivative are the key features of the present article.     

Dynamic programming strategy based on a type-2 fuzzy wavelet neural network

Nonlinear Dynamics - In this paper, an optimal control scheme, based on dynamic programming strategy, is presented for synchronization of uncertain fractional-order chaotic/hyperchaotic systems. In...     

Synchronization of piecewise continuous systems of fractional order

This paper proves analytically that synchronization of a class of piecewise continuous fractional-order systems can be achieved. Since there are no dedicated numerical methods to integrate differential equations with discontinuous right-hand sides for fractional-order models, Filippov’s regularization (Filippov, Differential Equations with Discontinuous Right-Hand Sides, 1988) is applied, and Cellina’s Theorem (Aubin and Cellina, Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska, Set-valued Analysis, 1990) is used. It is proved that the corresponding initial value problem can be converted to a continuous problem of fractional-order systems, to which numerical methods can be applied. In this way, the synchronization problem is transformed into a standard problem for continuous fractional-order systems. Three examples are presented: the Sprott’s system, Chen’s system, and Shimizu–Morioka’s system.     

Chaos in the fractional-order complex Lorenz system and its synchronization

In this article, a novel dynamic system, the fractional-order complex Lorenz system, is proposed. Dynamic behaviors of a fractional-order chaotic system in complex space are investigated for the first time. Chaotic regions and periodic windows are explored as well as different types of motion shown along the routes to chaos. Numerical experiments by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent are involved. A new method to search the lowest order of the fractional-order system is discussed. Based on the above result, a synchronization scheme in fractional-order complex Lorenz systems is presented and the corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

Hyperchaos synchronization of fractional-order arbitrary dimensional dynamical systems via modified sliding mode control

This paper considers the design of adaptive sliding mode control approach for synchronization of a class of fractional-order arbitrary dimensional hyperchaotic systems with unknown bounded disturbances. This approach is based on the principle of sliding mode control and adaptive compensation term for solving the problem of synchronization of the unknown parameters in fractional-order nonlinear systems. In particular, a novel fractional-order five dimensional hyperchaotic system has been introduced as a representative example. Furthermore, global stability and asymptotic synchronization between the outputs of master and slave systems can be achieved based on the modified Lyapunov functional and fractional stability condition. Simulation results are provided in detail to illustrate the performance of the proposed approach.     

Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes

A fractional-order weighted complex network consists of a number of nodes, which are the fractional-order chaotic systems, and weighted connections between the nodes. In this paper, we investigate generalized chaotic synchronization of the general fractional-order weighted complex dynamical networks with nonidentical nodes. The well-studied integer-order complex networks are the special cases of the fractional-order ones. Based on the stability theory of linear fraction-order systems, the nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are provided to verify the theoretical results. It is worth noting that the synchronization effect sensitively depends on both the fractional order ?? and the feedback gain k _i . Moreover, generalized synchronization of the fractional-order weighted networks can still be achieved effectively with the existence of noise perturbation.     

FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization

This paper introduces two novel fractional-order chaotic systems with cubic nonlinear resistor and investigates its adaptive sliding mode synchronization. Firstly the novel two equilibrium chaotic system with cubic nonlinear resistor (NCCNR) is derived and its dynamic properties are investigated. The fractional-order cubic nonlinear resistor system (FONCCNR) is then derived from the integer-order model and its stability and fractional-order bifurcation are discussed. Next a novel no-equilibrium chaotic cubic nonlinear resistor system (NECNR) is derived from NCCNR system. Dynamic properties of NECNR system are investigated. The fractional-order no equilibrium cubic nonlinear resistor system (FONECNR) is derived from NECNR. Stability and fractional-order bifurcation are investigated for the FONECNR system. The non-identical adaptive sliding mode synchronization of FONCCNR and FONECNR systems are achieved. Finally the proposed systems, adaptive control laws, sliding surfaces and adaptive controllers are implemented in FPGA.     

A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems

In this paper, we discuss and investigate the impulsive synchronization of fractional-order discrete-time chaotic systems. The proposed method is based on the impulsive synchronization theory used in the integer-order case on the one hand and the mathematical analysis of the fractional-order discrete-time systems on the other hand. Sufficient conditions for the stability of synchronization error system are given, and application example with numerical simulations is illustrated in order to verify that the proposed method is applicable and effective. Furthermore, in order to validate the proposed synchronization approach, we have also provided the experimental implementation results using Arduino Mega boards.     

Dynamics and synchronization analysis of coupled fractional-order nonlinear electromechanical systems

A fractional-order (FO) nonlinear model is used to describe an electromechanical system. We make capital out of the fact that for realistic modeling, the electric characteristics of a capacitor include a fractional-order time derivative. The dynamics and synchronization of coupled fractional-order nonlinear electromechanical systems are analyzed. Detailed attention is granted to the bifurcations that can occur in the dynamics of a single uncoupled electromechanical system as the fractional-order varies. For example, the fractional-order counterparts of the chaotic 4-order system are periodic at orders less than 3.985. The effect of the fractional-order on the condition of occurrence of synchronization phenomena in the network of many mutually coupled fractional-order nonlinear electromechanical systems is analyzed, especially when they are chaotic. An insight on the overall dynamics of the network is provided. It is shown that the dynamics of both the uncoupled system and the network are very sensitive to changes in the order of the fractional derivative.     

Nonlinear state-observer control for projective synchronization of a fractional-order hyperchaotic system

In this paper, we consider an observer-based control approach for manipulating projective synchronization of nonlinear systems in high dimensional. Based on the stability theory of the fractional-order dynamical system, a nonlinear state observer is designed which can achieve projective synchronization in a class of high dimensional fractional-order hyperchaotic systems without restriction of partial-linearity and calculating the Lyapunov index of system. Simulation studies are included to demonstrate the effectiveness and feasibility of the proposed approach and synthesis procedures.