Adaptive generalized function projective synchronization of uncertain chaotic complex systems

The complex nonlinear systems appear in many important fields of physics and engineering, which are very useful for cryptography and secure communication. This paper investigates adaptive generalized function projective synchronization (AGFPS) between two different dimensional chaotic complex systems with fully or partially unknown parameters via both reduced order and increased order. Based on the Lyapunov stability theorem and adaptive control technique, a general adaptive controller with corresponding parameter update rule is constructed to achieve AGFPS between two nonidentical chaotic complex systems with distinct orders, and identify the unknown parameters simultaneously. This scheme is then applied to obtain AGFPS between the hyperchaotic complex Lü system and the chaotic complex Lorenz system with fully unknown parameters, and between the uncertain chaotic complex Chen system and the uncertain hyperchaotic complex Lorenz system, respectively. Corresponding simulations results are performed to show the feasibility and effectiveness of the proposed synchronization method.     

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.     

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.     

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.     

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.     

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.     

Complete synchronization of delay hyperchaotic Lü system via a single linear input

This work is devoted to investigating the complete synchronization of two identical delay hyperchaotic Lü systems with different initial conditions, and a simple complete synchronization scheme only with a single linear input is proposed. Based on the Lyapunov stability theory, sufficient conditions of synchronization are obtained for both linear feedback and adaptive control approaches. The problem of adaptive synchronization between two nearly identical delay hyperchaotic Lü systems with unknown parameters is also studied. A?single input adaptive synchronization controller is proposed, and the adaptive parameter update laws are developed. Numerical simulation results are presented to demonstrate the effectiveness of the proposed chaos synchronization scheme.     

Lag synchronization of a class of chaotic systems with unknown parameters

Research on chaos synchronization of dynamical systems has been largely reported in literature. However, synchronization of different structure—uncertain dynamical systems—has received less attention. This paper addresses synchronization of a class of time-delay chaotic systems containing uncertain parameters. A unified scheme is established for synchronization between two strictly different time-delay uncertain chaotic systems. The synchronization is successfully achieved by designing an adaptive controller with the estimates of the unknown parameters and the nonlinear feedback gain. The result is rigorously proved by the Lyapunov stability theorem. Moreover, we illustrate the application of the proposed scheme by numerical simulation, which demonstrates the effectiveness and feasibility of the proposed synchronization method.     

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.     

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.     

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.     

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.     

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.     

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.     

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 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.

     

Full- and reduced-order synchronization of a class of time-varying systems containing uncertainties

Investigation on chaos synchronization of autonomous dynamical systems has been largely reported in the literature. However, synchronization of time-varying, or nonautonomous, uncertain dynamical systems has received less attention. The present contribution addresses full- and reduced-order synchronization of a class of nonlinear time-varying chaotic systems containing uncertain parameters. A unified framework is established for both the full-order synchronization between two completely identical time-varying uncertain systems and the reduced-order synchronization between two strictly different time-varying uncertain systems. The synchronization is successfully achieved by adjusting the determined algorithms for the estimates of unknown parameters and the linear feedback gain, which is rigorously proved by means of the Lyapunov stability theorem for nonautonomous differential equations together with Barbalat’s lemma. Moreover, the synchronization result is robust against the disturbance of noise. We illustrate the applicability for full-order synchronization using two identical parametrically driven pendulum oscillators and for reduced-order synchronization using the parametrically driven second-order pendulum oscillator and an additionally driven third-order Rossler oscillator.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Lag synchronization of hyperchaotic complex nonlinear systems

In this paper, we study the lag synchronization (LS) of n-dimensional hyperchaotic complex nonlinear systems. The idea of the nonlinear control technique based on the complex Lyapunov function with lag in time is used to propose a scheme to investigate LS of hyperchaotic attractors of these systems. Both complex Lyapunov and control functions are introduced. For illustration, the scheme is applied to two hyperchaotic complex Lorenz systems. The real and complex control functions are derived analytically to achieve LS and to show that the complex error dynamical systems are globally stable. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve LS of two identical hyperchaotic attractors.     

Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems

A scheme is designed to achieve phase synchronization (PS) and antiphase synchronization (APS) for an n-dimensional hyperchaotic complex nonlinear system. For this scheme, we have used the idea of an active control technique based on Lyapunov stability analysis to determine analytically the complex control functions which are needed to achieve PS and APS. We applied this scheme, as an example, to study PS and APS of hyperchaotic attractors of two identical hyperchaotic complex Lorenz systems. These complex systems appear in many important fields of physics and engineering. Our scheme can also be applied to two different hyperchaotic complex systems, for which PS and APS have not been investigated, as far as we know, in the literature. Numerical results are plotted to show phases and amplitudes of these hyperchaotic attractors, thus demonstrating that PS and APS are achieved. The bifurcation diagrams are computed for a wide range of parameters of the system parameters and are found to be symmetrical about the horizontal axis for APS, while they lack any symmetry for PS.