Robust adaptive full state hybrid synchronization of chaotic complex systems with unknown parameters and external disturbances

This paper studies the robust adaptive full state hybrid projective synchronization (FSHPS) scheme for a class of chaotic complex systems with uncertain parameters and external disturbances. By introducing a compensator and using nonlinear control and adaptive control, the robust adaptive FSHPS scheme is derived, which can eliminate the influence of uncertainties effectively and achieve adaptive FSHPS of the chaotic (hyperchaotic) complex systems asymptotically with a small error bound. The adaptive laws of the unknown parameters are given, and the sufficient conditions of realizing FSHPS are derived as well. Moreover, we also discuss the case that parameters of chaotic complex system are complex. Finally, the complex Chen system and Lü system, and the hyperchaotic complex Lorenz system are taken as two examples and the numerical simulations are provided to verify the effectiveness and robustness of the proposed control scheme.     

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.

     

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.     

Generalized projective lag synchronization between?different hyperchaotic systems with uncertain parameters

Generalized projective lag synchronization (GPLS) is characterized by the output of the drive system proportionally lagging behind the output of the response system. In this paper, GPLS between different hyperchaotic systems with uncertain parameters, i.e., GPLS between Lorenz and Lü hyperchaotic systems, and between Lorenz?CStenflo and Lorenz hyperchaotic systems, is studied by applying an adaptive control method. Based on Lyapunov stability theory, the adaptive controllers and corresponding parameter update rules are constructed to make the states of two diverse hyperchaotic systems asymptotically synchronize up to the desired scaling matrix and to estimate the uncertain parameters. Some numerical simulations are provided to show the effectiveness of our results.     

A single adaptive controller with one variable for synchronizing two identical time delay hyperchaotic Lorenz systems with mismatched parameters

Time delays are ubiquitous in real world and are often sources of complex behaviors of dynamical systems. This paper addresses the problem of parameter identification and synchronization of uncertain hyperchaotic time-delayed systems. Based on the Lyapunov stability theory and the adaptive control theory, a single adaptive controller with one variable for synchronizing two identical time-delay hyperchaotic Lorenz systems with mismatch parameters is proposed. The parameter update laws and sufficient conditions of the scheme are obtained for both linear feedback and adaptive control approaches. Numerical simulations are also given to show the effectiveness of the proposed method.     

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.     

Adaptive modified function projective synchronization and parameter identification of uncertain hyperchaotic (chaotic) systems with identical or non-identical structures

Modified function projective synchronization (MFPS), which generalizes many kinds of synchronization form, has received great attention recently. Based on the active control method and adaptive control technique, a general formula for designing the controllers is proposed to achieve adaptive MFPS, which corrects several incomplete results that have been reported recently. In addition, this paper derives the sufficient condition for parameter identification, which was not mentioned in much of the relevant literature concerning MFPS. Furthermore, we extend the MFPS scheme to the cases that the drive and response systems come with non-identical structures. The proposed method is both theoretically rigorous and practically feasible, which has the merits that it can not only achieve the full-state MFPS but also identify the fully unknown parameters in the synchronization process. The theoretical results are successfully applied to three typical illustrative cases: the adaptive MFPS of two identical 4-D hyperchaotic systems with unknown parameters in the response system, the adaptive MFPS between a 5-D hyperchaotic system and a 4-D hyperchaotic system with unknown parameters in the drive system and the adaptive MFPS between a 3-D chaotic system and a 4-D hyperchaotic system when the parameters in the drive system and response system are all unknown. For each case the controller functions and parameter update laws are well designed in detail. Moreover, the corresponding numerical simulations are presented, which agree well with the theoretical analysis.     

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.     

The alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay

The various cases of synchronization in two identical hyperchaotic Lorenz systems with time delay are studied. Based on Lyapunov stability theory, the sufficient conditions for achieving synchronization of two identical hyperchaotic Lorenz systems with time delay are derived, and a simple scheme only with a single linear controller is proposed. When the parameters in the response system are known, the alternating between complete synchronization and hybrid synchronization (namely, coexistence of antiphase and complete synchronization) is observed with the control feedback gain varying. Furthermore, when the parameters in the response system are unknown, for the same feedback controller, the complete synchronization and the hybrid synchronization can be obtained, respectively, as the associated parameters updated laws of the unknown parameters are chosen. Numerical simulation results are presented to demonstrate the proposed chaos synchronization scheme.     

Projective synchronization of hyperchaotic Lü system and Liu system

This work is concerned with projective synchronization of hyperchaotic Lü system and Liu system by add-order method. Different controllers are designed to projective-synchronize the two nonidentical chaotic systems, active control is used when parameters are known, while the adaptive control law and the parameter update rule are derived via adaptive control when parameters are uncertain. Moreover, the convergence rates of the scheme can be adjusted by changing the control coefficients. Finally, numerical simulations are also shown to verify the results.     

Adaptive synchronization of Lü hyperchaotic system with uncertain parameters based on single-input controller

The adaptive synchronized problem of the four-dimensional (4D) Lü hyperchaotic system performed by Elabbasy et al. (Chaos Solitons Fractals 30:1133–1142, 2006) with uncertain parameters by applying the single control input is addressed in this article. Based on the Lyapunov theorem of stability, the single-input adaptive synchronization controllers associated with the adaptive update laws of system parameters are developed to make the states of two nearly identical 4D Lü hyperchaotic systems asymptotically synchronized. Numerical studies are presented to illustrate the effectiveness of the proposed chaotic synchronization schemes.     

Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.     

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.     

Projective synchronization of new hyperchaotic system with fully unknown parameters

Projective synchronization of new hyperchaotic Newton–Leipnik system with fully unknown parameters is investigated in this paper. Based on Lyapunov stability theory, a new adaptive controller with parameter update law is designed to projective synchronize between two hyperchaotic systems asymptotically and globally. Basic bifurcation analysis of the new system is investigated by means of Lyapunov exponent spectrum and bifurcation diagrams. It is found that the new hyperchaotic system possesses two positive Lyapunov exponents within a wide range of parameters. Numerical simulations on the hyperchaotic Newton–Leipnik system are used to verify the theoretical results.     

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 general nonlinear adaptive control scheme for finite-time synchronization of chaotic systems with uncertain parameters and nonlinear inputs

In this paper, the problem of finite-time chaos synchronization between two different uncertain chaotic systems with unknown parameters and input nonlinearities is investigated. It is assumed that both master and slave systems are perturbed by unknown model uncertainties, external disturbances, and fully unknown parameters. Proper update laws are proposed to estimate the systems?? unknown parameters. Based on the update laws and finite-time control technique, a robust adaptive controller is introduced to guarantee the convergence of the slave system trajectories to the trajectories of the master system in a given finite time. Two illustrative examples are presented to illustrate the effectiveness and applicability of the proposed finite-time controller and to validate the theoretical results of the paper.     

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.     

Robust adaptive synchronization for a class of chaotic systems with actuator failures and nonlinear uncertainty

This paper is concerned with the robust adaptive synchronization problem for a class of chaotic systems with actuator failures and unknown nonlinear uncertainty. Combining adaptive method and linear matrix inequality (LMI) technique, a novel type of robust adaptive reliable synchronization controller is proposed, which can eliminate the effect of actuator fault and nonlinear uncertainty on systems. After solving a set of LMIs, synchronization error between the master chaotic and slave chaotic systems can converge asymptotically to zero. Finally, illustrate examples about chaotic Chua’s circuit system and Lorenz systems are provided to demonstrate the effectiveness and applicability of the proposed design method.     

Generalized Darboux transformation and parameter-dependent rogue wave solutions to a nonlinear Schrödinger system

Objectives of the paper are (1) to design two new real and complex no equilibrium point hyperchaotic systems, (2) to design synchronisation technique for the new systems using the contraction theory and (3) to validate the results by using circuit realisation. First a new no equilibrium point hyperchaotic system is developed using a 3-D generalised Lorenz system; then using the new system a new complex no equilibrium point hyperchaotic system is reported. Both the new systems have hidden chaotic attractors. Various dynamical behaviours are observed in the new systems like chaotic, periodic, quasi-periodic and hyperchaotic. Both the systems have inverse crisis route to chaos with the variation of parameter a and crisis route to chaos with the variation of parameters \(b,\ c\) and d. These phenomena along with hidden attractors in a complex hyperchaotic system are not seen in the literature. Synchronisation between the identical new hyperchaotic systems is achieved using the contraction theory. Further the synchronisation between the identical new complex hyperchaotic systems is achieved using adaptive contraction theory. The proposed synchronisation strategies are validated using the MATLAB simulation and circuit implementation results. Further, an application of the proposed system is shown by transmitting and receiving an audio signal.     

Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters

In this paper, the synchronization of fractional order chaotic systems with random and uncertain parameters is analyzed. Firstly, based on the orthogonal polynomial expansion, the fractional order Lü and Lorenz systems with random and uncertain parameters are reduced into the equivalent deterministic systems. Secondly, modified projective synchronization of equivalent deterministic Lü and Lorenz systems is explored. Lastly, the theoretical results are verified by the numerical simulations.