Projective synchronization between two different time-delayed chaotic systems using active control approach

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.     

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.     

Complex projective synchronization in coupled chaotic complex dynamical systems

In previous papers, the projective factors are always chosen as real numbers, real matrices, or even real-valued functions, which means the coupled systems evolve in the same or inverse direction simultaneously. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, the projective synchronization with respect to a complex factor, called complex projective synchronization (CPS), should be taken into consideration. In this paper, based on Lyapunov stability theory, three typical chaotic complex dynamical systems are considered and the corresponding controllers are designed to achieve the complex projective synchronization. Further, an adaptive control method is adopted to design a universal controller for partially linear systems. Numerical examples are provided to show the effectiveness of the proposed method.     

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.     

Analyzing projective synchronization on different scaling factors in a drive-response coupling dynamical network with time-varying delays

In this paper, projective synchronization of drive-response coupled dynamical network with delayed system nodes and coupling time-varying delay is investigated via impulsive control, where the scaling factors are different from each other. Different controllers are designed to achieve the projective synchronization: only impulsive control is used when the scaling factors need extra limitation, while an extra controller, that is, a simple linear feedback controller, is added when the scaling factors don??t need extra limitation. Based on the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization of such coupled network are established, and an estimate of the upper bound of impulsive intervals ensuring global exponential synchronization of drive-response coupled dynamical network is also given. Numerical examples on the time-delay Lorenz chaotic systems are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Hybrid robust modified function projective lag synchronization in two different dimensional chaotic systems

In this article, a novel synchronization scheme, modified function projective lag synchronization (MFPLS) in two different dimensional chaotic systems with parameter perturbations, is proposed. In the proposed method, the states of two nonidentical chaotic systems with different orders are asymptotically lag synchronized up to a desired scaling function matrix by means of reduced order and increased order, respectively. Furthermore, based on the reality situation, the parameter perturbations are involved, which are assumed to appear in both drive and response systems. With the Lyapunov stability theory, an adaptive controller is designed to achieve MFPLS. Theoretical proof and numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

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.     

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have non-identical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to assess the performance of the proposed adaptive controller in synchronizing chaotic systems.     

Modified projective and modified function projective synchronization of a class of real nonlinear systems and a class of complex nonlinear systems

The modified projective and modified function projective synchronization of a class of chaotic real nonlinear systems, or a class of chaotic complex nonlinear systems, have been widely reported in the previous studies, respectively. In the paper, the modified projective and modified function projective synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems are first investigated. Based on the Lyapunov stability theory, the drive real system and response complex system can be synchronized up to the desired scaling constants and functions, respectively. The corresponding numerical simulations are performed to verify and demonstrate the validity and feasibility of the presented idea.     

Mean square function synchronization of chaotic systems with stochastic effects

In this paper, we give the definition of mean square function synchronization. Secondly, we investigate mean square function synchronization of chaotic systems with stochastic perturbation and unknown parameters. Based on the Lyapunov stability theory, inequality techniques, and the properties of the Weiner process, the controller, and adaptive laws are designed to ensure achieving stochastic synchronization of chaotic systems. A sufficient synchronization condition is given to ensure the chaotic systems to be mean-square stable. Furthermore, a numerical simulation is also given to demonstrate the effectiveness of the proposed scheme.     

Combination–combination synchronization among four identical or different chaotic systems

Based on one drive system and one response system synchronization model, a new type of combination–combination synchronization is proposed for four identical or different chaotic systems. According to the Lyapunov stability theorem and adaptive control, numerical simulations for four identical or different chaotic systems with different initial conditions are discussed to show the effectiveness of the proposed method. Synchronization about combination of two drive systems and combination of two response systems is the main contribution of this paper, which can be extended to three or more chaotic systems. A universal combination of drive systems and response systems model and a universal adaptive controller may be designed to our intelligent application by our synchronization design.     

Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control

In this paper, we apply the nonsingular terminal sliding mode control technique to realize the novel combination-combination synchronization between combination of two chaotic systems as drive system and combination of two chaotic systems as response system with unknown parameters in a finite time. On the basic of the adaptive laws and finite-time stability theory, an adaptive combination sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time for four different chaotic systems. In theory, it is proved that the sliding mode technique can realize fast convergence for four different chaotic systems in the finite time. Some criteria and corollaries are derived for finite-time combination-combination synchronization of four different chaotic systems. Numerical simulation results are shown to verify the effectiveness and correctness of the combination-combination synchronization.     

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.     

Projective synchronization of the small-world delayed network with uncertainty

Spatiotemporal chaotic systems are taken as nodes to constitute the small-world delayed network with uncertainty, and projective synchronization of the network is researched. The control input of the network and the adaptive law of adjustment parameters are designed based on Lyapunov theorem. Concretely, the Burgers systems with spatiotemporal chaotic behavior in physics domain are taken as nodes to constitute the complex network, and the Fisher–Kolmogorov system is taken as the target system. The simulation results show that the synchronization performance of the network is very stable.     

A kind of binary scaling function projective lag synchronization of chaotic systems with stochastic perturbation

This paper investigates a kind of binary function projective lag synchronization of uncertain chaotic systems with stochastic perturbation. In comparison with those of the existing scaling function synchronization, we assume that the given scaling function can be the binary boundary function, even an n-ary boundary function. Based on the LaSalle-type invariance principle for stochastic differential equation, the adaptive control law is derived to make the state of two chaotic systems function projective lag synchronized. Some numerical is also given to show the effectiveness of the proposed method.     

Complex projective synchronization in drive-response networks coupled with complex-variable chaotic systems

In drive-response complex-variable systems, projective synchronization with respect to a real number, real matrix, or even real function means that drive-response systems evolve simultaneously along the same or inverse direction in a complex plane. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, this paper investigates projective synchronization in drive-response networks of coupled complex-variable chaotic systems with respect to complex numbers, called complex projective synchronization (CPS). The adaptive feedback control method is adopted first to achieve CPS in a general drive-response network. For a special class of drive-response networks, the CPS is achieved via pinning control. Furthermore, a universal pinning control scheme is proposed via the adaptive coupling strength method, several simple and useful criteria for CPS are obtained, and all results are illustrated by numerical examples.     

Projective synchronization of a class of chaotic systems by dynamic feedback control method

This paper investigates the projective synchronization problem of a class of chaotic systems in arbitrary dimensions. Firstly, a necessary and sufficient condition for the existence of the projective synchronization problem is presented. And this condition is equivalent to check whether a group of algebraic equations about \(\alpha \) have solutions or not. Secondly, an algorithm is proposed to obtain all the solutions of the projective synchronization problem. Thirdly, a simple and physically implementable controller is designed to ensure the realization of the projective synchronization. Finally, three numerical examples are provided to verify the effectiveness and the validity of the proposed results.     

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.     

Chaos control and modified projective synchronization of unknown heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive backstepping control

This paper proposes the chaos control and the modified projective synchronization methods for unknown heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive backstepping control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a regular or periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise-like, synchronized chaotic systems can be used as cipher generators for secure communication. Obviously, the importance of obtaining these objectives is specified when the dynamics of gyroscope system are unknown. In this paper, using the neural backstepping control technique, control laws are established which guarantees the chaos control and the modified projective synchronization of unknown chaotic gyroscope system. In the neural backstepping control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimators are derived in the sense of Lyapunov function. Thus, the unknown chaotic gyroscope system can be guaranteed to be asymptotically stable. Also, the control objectives have been achieved.     

Modified projective synchronization of fractional-order chaotic systems via active sliding mode control

This paper is devoted to study the problem of modified projective synchronization of fractional-order chaotic system. Base on the stability theorems of fractional-order linear system, active sliding mode controller is proposed to synchronize two different fractional-order systems. Moreover, the controller is robust to the bounded noise. Numerical simulations are provided to show the effectiveness of the analytical results.