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.     

A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov??s and Shilnikov??s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use improved version of the Adams?CBashforth?CMoulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order q=0.9, from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order q=0.9. An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.     

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.     

Complete synchronization of chaotic complex nonlinear systems with uncertain parameters

Our main objective in this work is to investigate complete synchronization (CS) of n-dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.     

Observer-based synchronization in fractional-order leader–follower complex networks

This paper is devoted to synchronization behavior of complex dynamical networks with a Caputo fractional-order derivative. In particular, we propose a fractional-order leader–follower complex network where the leader is independent, has its own dynamics, and is followed by all the other nodes. Using the state observer approach, the leader and the followers are designed to connect through only a scalar coupling signal instead of the commonly used full state coupling scheme. On the basis of stability theory of the fractional-order differential system, a sufficient condition for global network synchronization is presented, which only relates to the linear part of individual nodes and can be easily solved by the pole-placement design technique. The analytic results are complemented with numerical simulations for a network whose nodes are governed by the fractional-order Chua’s circuit.     

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.     

Dynamics analysis and hybrid function projective synchronization of a new chaotic system

This paper introduces a novel three-dimensional autonomous chaotic system by adding a quadratic cross-product term to the first equation and modifying the state variable in the third equation of a chaotic system proposed by Cai et al. (Acta Phys. Sin. 56:6230, 2007). By means of theoretical analysis and computer simulations, some basic dynamical properties, such as Lyapunov exponent spectrum, bifurcations, equilibria, and chaotic dynamical behaviors of the new chaotic system are investigated. Furthermore, hybrid function projective synchronization (HFPS) of the new chaotic system is studied by employing three different synchronization methods, i.e., adaptive control, system coupling and active control. The proposed approaches are applied to achieve HFPS between two identical new chaotic systems with fully uncertain parameters, HFPS in coupled new chaotic systems, and HFPS between the integer-order new chaotic system and the fractional-order Lü chaotic system, respectively. Corresponding numerical simulations are provided to validate and illustrate the analytical results.     

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.     

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.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

Generalized finite-time synchronization between coupled chaotic systems of different orders with unknown parameters

This letter investigates the adaptive finite-time synchronization of different coupled chaotic (or hyperchaotic) systems with unknown parameters. The sufficient conditions for achieving the generalized finite-time synchronization of two chaotic systems are derived based on the theory of finite-time stability of dynamical systems. By the adaptive control technique, the control laws and the corresponding parameters update laws are proposed such that the generalized finite-time synchronization of nonidentical chaotic (or hyperchaotic) systems is to be obtained. These results obtained are in good agreement with the existing one in open literature and it is shown that the technique introduced here can be further applied to various finite-time synchronizations between dynamical systems. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed scheme.     

Bursting generation mechanism in a three-dimensional autonomous system,chaos control,and synchronization in its fractional-order form

The bifurcation mechanism of bursting oscillations in a three-dimensional autonomous slow-fast Kingni et al. system (Nonlinear Dyn. 73, 1111–1123, 2013) and its fractional-order form are investigated in this paper. The stability analysis of the system is carried out assuming that the slow subsystem evolves on quasi-static state. It is reveaved that the bursting oscillations found in the system result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. We refer this class of bursting to “source/bursting.” The coexistence of symmetrical bursting limit cycles and chaotic bursting attractors is observed. In addition, the fractional-order chaotic slow-fast system is studied. The lowest order of the commensurate form of this system to exhibit chaotic behavior is found to be 2.199. By tuning the commensurate fractional-order, the chaotic slow-fast system displays Chen- and Lorenz-like chaotic attractors, respectively. The stability analysis of the controlled fractional-order-form of the system to its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Moreover, the synchronization of chaotic bursting oscillations in two identical fractional-order systems is numerically studied using the unidirectional linear error feedback coupling scheme. It is shown that the system can achieve synchronization for appropriate coupling strength. Furthermore, the effect of fractional derivatives orders on chaos control and synchronization is analyzed.     

Synchronization in a fractional-order dynamic network with uncertain parameters using an adaptive control strategy

This paper studies synchronization of all nodes in a fractional-order complex dynamic network. An adaptive control strategy for synchronizing a dynamic network is proposed. Based on the Lyapunov stability theory, this paper shows that tracking errors of all nodes in a fractional-order complex network converge to zero. This simple yet practical scheme can be used in many networks such as small-world networks and scale-free networks. Unlike the existing methods which assume the coupling configuration among the nodes of the network with diffusivity, symmetry, balance, or irreducibility, in this case, these assumptions are unnecessary, and the proposed adaptive strategy is more feasible. Two examples are presented to illustrate effectiveness of the proposed method.     

Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach

In this paper, an adaptive sliding mode control method is introduced to ensure robust synchronization of two different fractional-order chaotic systems with fully unknown parameters and external disturbances. For this purpose, a fractional integral sliding surface is defined and an adaptive sliding mode controller is designed. In this method, no knowledge of the bounds of parameters and perturbation is required in advance and the parameters are updated through an adaptive control process. The proposed scheme is global and theoretically rigorous. Two examples are given to illustrate effectiveness of the scheme, in which the synchronizations between fractional-order chaotic Chen system and fractional-order chaotic Rössler system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.

     

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.     

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.     

Finite-time generalized synchronization of chaotic systems with different order

In this paper, the generalized synchronization of chaotic systems with different order is studied. The definition of finite-time generalized synchronization is put forward for the first time. Based on the finite-time stability theory, two control strategies are proposed to realize the generalized synchronization of chaotic systems with different order in finite time. Besides the relation between the parameter β, the initial states of systems and the convergent time were obtained. The corresponding numerical simulations are presented to demonstrate the effectiveness of proposed schemes.     

Adaptive generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimensions

Generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimension of nodes via adaptive control method is investigated in this paper. Based on Lyapunov stability theory, adaptive controller is obtained and unknown parameters of both the drive network and the response network are estimated by adaptive laws. In addition, the three-dimension chaotic system and the four-dimension hyperchaotic system, respectively, as the nodes of the drive and response network are analyzed in detail, and numerical simulation results are presented to illustrate the effectiveness of the theoretical results.     

Parameter identification and adaptive impulsive synchronization of uncertain complex-variable chaotic systems

Synchronization of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this paper, the problem of parameter identification and adaptive impulsive synchronization for a class of chaotic (hyperchaotic) complex nonlinear systems with uncertain parameters is investigated. Based on the theories of adaptive control and impulsive control, a synchronization scheme is designed to make a class of chaotic and hyperchaotic complex systems asymptotically synchronized, and uncertain parameters are identified simultaneously in the process of synchronization. Particularly, the proposed adaptive–impulsive control laws for synchronization are simple and can be readily applied in practical applications. The synchronization of two identical chaotic complex Chen systems and two identical hyperchaotic complex Lü systems are taken as two examples to verify the feasibility and effectiveness of the proposed controllers and identifiers.     

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.