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.     

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.     

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.     

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.     

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.     

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.

     

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.     

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.     

Generalized synchronization of fractional-order hyperchaotic systems and its DSP implementation

In this paper, we investigate synchronization and its DSP implementation of fractional-order simplified Lorenz hyperchaotic systems by employing the Adomian decomposition method. The active controller and linear feedback controller are designed. Numerical simulation of the synchronized systems is carried out, and it is found that the synchronization phenomenon can be observed in both state variables and intermediate variables. Moreover, the synchronized systems are implemented in two TMS320F2-8335 DSP boards which are connected by a serial port and the output signals are exhibited by an oscilloscope. The experiment results show that the proposed implementation method works well on DSP.     

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.     

Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters via nonlinear control

A new adaptive synchronization scheme by pragmatical asymptotically stability theorem is proposed in this paper. Based on this theorem and nonlinear control theory, a new adaptive synchronization scheme to design controllers can be obtained and especially the constraints for minimum values of feedback gain K in controllers can be derived. This new strategy shows that the constraint values of feedback gain K are related to the error of unknown and estimated parameters if the goal system is given. Through this new strategy, an appropriate feedback gain K can be always decided easily to obtain controllers achieving adaptive synchronization. Two identical Lorenz systems with different initial conditions and two completely different nonlinear systems with different orders, augmented R?ssler??s system and Mathieu?Cvan der Pol system, are used for illustrations to demonstrate the efficiency and effectiveness of the new adaptive scheme in numerical simulation results.     

Combination synchronization of three different order nonlinear systems using active backstepping design

In this paper, two kinds of combination synchronization between two drive systems and one response system are investigated using active backstepping design. Firstly, increased-order combination synchronization between Lorenz system, Rössler system and hyperchaotic Lü system is considered. Secondly, reduced-order combination synchronization between hyperchaotic Lorenz system, hyperchaotic Chen system and Lü system is considered. According to Lyapunov stability theory and active backstepping design method, the corresponding controllers are both designed. Finally, several numerical examples are provided to illustrate the obtained results.     

Study on synchronization and parameters insensitivity of a class of hyperchaotic systems using nonlinear feedback control

A class of hyperchaotic systems has strong noise robustness. When conventional synchronization algorithms are used in this system, however, the convergence rate of synchronization is slow, and the synchronization performances are very sensitive to the parameters of response system. To resolve the problems, synchronization using nonlinear feedback control is proposed. According to Hurwitz stability theory, designing a nonlinear controller can make the real parts of the eigenvalues of the error equation’s Jacobian matrix negative. And the absolute values of the eigenvalues are larger, the convergence rate of synchronization is faster. Besides, the theoretical results of parameters insensitivity are given. Finally, numerical simulations are given to verify and test the correctness and effectiveness of the methods we proposed.     

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.     

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.     

Robust synchronization of Rossler systems with mismatched time-varying parameters

This paper presents robust synchronization algorithms for the Rossler systems in the presence of unknown time-varying parameters. First, an adaptive synchronization algorithm based on the Lyapunov theory is introduced for identical Rossler systems with mismatched uncertainties. This method does not require a priori information regarding the bound of uncertainties. In addition, this technique is such that the states of the synchronization error system are uniformly ultimately bounded. Since in practice the parameters of the drive and response systems are not necessarily the same, two synchronization approaches are used for the drive and response systems with different parameters. In the first approach, a simple controller is designed for the nominal error system, as if there is no uncertainty in the system. The stability analysis is then investigated as the uncertainties are reintroduced, and it is shown that the size of the uncertainties directly affects the synchronization performance. To deal with this problem, an H _∞ controller is designed in which the effects of unknown bounded uncertainties can be attenuated at an appropriate level. It is shown that, using these two approaches, the Rossler systems can be synchronized effectively and the synchronization error is uniformly ultimately bounded. Numerical simulations confirm the effectiveness of the proposed methods.     

Chaos bursting synchronization of mismatched Hindmarsh–Rose systems via a single adaptive feedback controller

Chaotic bursting synchronization of mismatched Hindmarsh–Rose neuron systems is investigated. Based on the Lyapunov stability theory, an adaptive feedback control scheme for the synchronization of the neuron systems is proposed when partially parameters of the response system are unknown and different with those of the drive system. Furthermore, in the proposed scheme, only a single adaptive feedback controller is needed, which is efficient and easy to implement. Finally, numerical simulations are provided to show the effectiveness of the developed methods.     

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.     

Analog circuit design and optimal synchronization of a modified Rayleigh system

This paper addresses the problem of optimization of the synchronization of a chaotic modified Rayleigh system. We first introduce a four-dimensional autonomous chaotic system which is obtained by the modification of a two-dimensional Rayleigh system. Some basic dynamical properties and behaviors of this system are investigated. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the proposed system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. Furthermore, we propose an optimal robust adaptive feedback which accomplishes the synchronization of two modified Rayleigh systems using the controllability functions method. The advantage of the proposed scheme is that it takes into account the energy wasted by feedback coupling and the closed loop performance on synchronization. Also, a finite horizon is explicitly computed such that the chaos synchronization is achieved at an established time. Numerical simulations are presented to verify the effectiveness of the proposed synchronization strategy. Pspice analog circuit implementation of the complete master–slave controller system is also presented to show the feasibility of the proposed scheme.     

Spatiotemporal combination synchronization of different nonlinear objects

This paper proposes a novel secure communication scheme based on the Karhunen–Loéve decomposition and the synchronization of a master and a slave hyperchaotic Lü systems. First, the Karhunen–Loéve decomposition is used as a data reduction tool to generate data coefficients and eigenfunctions that capture the essence of grayscale and color images in an optimal manner. It is noted that the original images can be reproduced using only the most energetic eigenfunctions; this results in computational savings. The data coefficients are encrypted and transmitted using a master hyperchaotic Lü system. These coefficients are then recovered at the receiver end using a sliding mode controller to synchronize two hyperchaotic Lü systems. Simulation results are presented to illustrate the ability of the proposed control law to synchronize the master and slave hyperchaotic Lü systems. Moreover, the original images are recovered by using the decrypted data coefficients in conjunction with the eigenfunctions of the image. Computer simulation results are provided to show the excellent performance of the proposed scheme.