Adaptive Control of Nonlinear Dynamical Systems Using a Model Reference Approach

In this paper we consider using a model reference adaptive control approach to control nonlinear systems. We consider the controller design and stability analysis associated with these type of adaptive systems. Then we discuss the use of model reference adaptive control algorithms to control systems which exhibit nonlinear dynamical behaviour using the example of a Duffing oscillator being controlled to follow a linear reference model. For this system we show that if the nonlinearity is small then standard linear model reference control can be applied. A second example, which is often found in synchronization applications, is when the nonlinearities in the plant and reference model are identical. Again we show that linear model reference adaptive control is sufficient to control the system. Finally we consider controlling more general nonlinear systems using adaptive feedback linearization to control scalar nonlinear systems. As an example we use the Lorenz and Chua systems with parameter values such that they both have chaotic dynamics. The Lorenz system is used as a reference model and a single coordinate from the Chua system is controlled to follow one of the Lorenz system coordinates.     

Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking

We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain variable parameters by linear coupling and pragmatical adaptive tracking. The uncertain parameters of a system vary with time due to aging, environment, and disturbances. A?sufficient condition is given for the asymptotical stability of common zero solution of error dynamics and parameter update dynamics by the Ge?CYu?CChen pragmatical asymptotical stability theorem based on equal probability assumption. Numerical results are studied for a Lorenz system and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchronization strategy.     

Synchronization of stochastic reaction–diffusion systems via boundary control

This paper studies the problem of mean square asymptotical synchronization and \(H_\infty \) synchronization for coupled stochastic reaction–diffusion systems (SRDSs) via boundary control. Based on the deduced synchronization error dynamic, we design boundary controllers to achieve mean square asymptotical synchronization. By virtue of Lyapunov functional method and Wirtinger’s inequality, sufficient conditions are obtained for ensuring mean square asymptotical synchronization. When coupled SRDSs are subject to external disturbance, mean square \(H_\infty \) synchronization is investigated and corresponding criterion is presented under a designed boundary controller. In addition to focusing on systems with Neumann boundary conditions, we also briefly study coupled SRDSs with mixed boundary conditions and sufficient conditions are provided to achieve the desired performance. Numerical examples are used to verify the effectiveness of our theoretical results.     

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.     

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.     

Synchronization and chaos control by quorum sensing mechanism

Diverse rhythms are generated by thousands of oscillators that somehow manage to operate synchronously. By using mathematical and computational modeling, we consider the synchronization and chaos control among chaotic oscillators coupled indirectly but through a quorum sensing mechanism. Some sufficient criteria for synchronization under quorum sensing are given based on traditional Lyapunov function method. The Melnikov function method is used to theoretically explain how to suppress chaotic Lorenz systems to different types of periodic oscillators in quorum sensing mechanics. Numerical studies for classical Lorenz and Rössler systems illustrate the theoretical results.     

Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems

This paper studies the generalized Lorenz canonical form of dynamical systems introduced by elikovský and Chen [International Journal of Bifurcation and Chaos 12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The ilnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have ilnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of ilnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.     

Characterizing the anticipating chaotic synchronization of RCL-shunted Josephson junctions

This study addresses the problem of anticipating synchronization of two chaotic RCL-shunted Josephson junctions (RCLSJ) based on time-delayed feedback control. The error dynamical system can be dealt with in virtue of non-linear pendulum-like system methods, through which we establish sufficient conditions to guarantee the existence of anticipating synchronizing slave systems. The design of a desired feedback controller can be achieved by solving a group of linear matrix inequalities (LMIs) by utilizing available numerical software. In the presence of parameter uncertainties, we further explore the robust anticipating synchronization, and propose corresponding criteria as well. These results are demonstrated through numerical simulations that under the derived conditions, the slave RCLSJ model could respond in exactly the same way as the master would do in the future, hence it allow us to anticipate the non-linear chaotic dynamics.     

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.     

$$H_\infty $$ switching synchronization for multiple time-delay chaotic systems subject to controller failure and its application to aperiodically intermittent control

This paper is concerned with the \(H_\infty \) synchronization control problem for a class of chaotic systems with multiple delays in the presence of controller temporary failure. Based on the idea of switching, the synchronization error systems with controller temporary failure are modeled as a class of switched synchronization error systems. Then, a switching condition that incorporates the controller failure time is constructed by using piecewise Lyapunov functional and average dwell-time methods, such that the switched synchronization error systems are exponentially stable and satisfy a weighted \(H_\infty \) performance level. In the meantime, a switching state feedback \(H_\infty \) controller is derived by solving a set of linear matrix inequalities. More incisively, the obtained results can also be applied to the issue of aperiodically intermittent control. Finally, three simulation examples are employed to illustrate the effectiveness and potential of the proposed method.     

Double-compound synchronization of six memristor-based Lorenz systems

In this paper, a new type of double-compound synchronization, which is based on combination–combination synchronization and compound synchronization of four chaotic systems, is investigated for six memristor-based Lorenz systems. Using Lyapunov stability theory and adaptive control, some sufficient conditions are attained to ensure our conclusions hold. The corresponding theoretical proofs and numerical simulations are supplied to verify the effectiveness and feasibility of our synchronization design. Due to the complexity of our synchronization, it will be more secure to transmit and receive signals in application of communication.     

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.     

Global synchronization criteria for two Lorenz–Stenflo systems via single-variable substitution control

The global synchronization of chaotic Lorenz–Stenflo systems via variable substitution control is studied. First, a master-slave synchronization scheme with variable substitution control is constructed. Based on this scheme, some sufficient criteria for the global chaos synchronization of master and slave Lorenz–Stenflo systems via various single-variable coupling are derived and formulated in the form of algebra. Numerical examples are provided to verify the effectiveness of the criteria.     

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.     

Lag quasisynchronization of coupled delayed systems with parameter mismatch by periodically intermittent control

This paper further investigates the lag synchronization of coupled delayed systems with parameter mismatch. Different from the most existing results, we formulate the intermittent control system that governs the dynamics of the synchronization error. As a result of parameter mismatch, complete lag synchronization cannot be achieved. In this paper, a lag quasisynchronization scheme is proposed to ensure that coupled systems are in a state of lag synchronization with an error level. We estimate the error bound of lag synchronization by rigorously theoretical analysis. Numerical simulations are presented to verify the theoretical results.     

Outer synchronization of uncertain small-world networks via adaptive sliding mode control

The outer synchronization of irregular coupled complex networks is investigated with nonidentical topological structures. The switching gain is estimated by an adaptive technique, and a sliding mode controller is designed to satisfy the sliding condition. The outer synchronization between two irregular coupled complex networks with different initial conditions is implemented via the designed controllers with the corresponding parameter update laws. The chaos synchronization of two small-world networks consisting of N uncertain identical Lorenz systems is achieved to demonstrate the applications of the proposed approach.     

Existence of a Singularly Degenerate Heteroclinic Cycle in the Lorenz System and Its Dynamical Consequences: Part I

We prove that the Lorenz system with appropriate choice of parameter values has a specific type of heteroclinic cycle, called a singularly degenerate heteroclinic cycle, that consists of a line of equilibria together with a heteroclinic orbit connecting two of the equilibria. By an arbitrarily small but carefully chosen perturbation to the Lorenz system, we also show that the geometric model of Lorenz attractors formulated by Guckenheimer will bifurcate from it, among other things. Although not proven, one may also expect various other types of chaotic dynamics such as Hénon-like chaotic attractors, Lorenz attractors with hooks which were recently studied by S. Luzzatto and M. Viana [22], and what were observed in the original Lorenz system with large r and small b in the Sparrows book [34]. Our analysis is all done within a family of three dimensional ODEs that contains, as its subfamilies, the Lorenz system, the Rösslers second system and the Shimizu–Morioka system, which are known to exhibit Lorenz-like chaotic dynamics.     

On the global asymptotic stability of periodic solutions of a class of ecological system

In this paper, we have made reaearches on the mathematical models which have three populations of mutual action: We have obtained the sufficient conditions respectively for the systems(*)and(**)for existence and uniqueness of single positive periodic solutions which are globally asymptotically stable.     

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.     

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.