Synchronization Of Time-delayed Systems Research Articles

Intermittent behavior near the boundary of generalized synchronization in unidirectionally coupled time-delayed systems

The aim of the work is an analysis of characteristics of intermittent behavior taking place near the boundary of generalized synchronization in unidirectionally coupled time-delayed generators. The case of interaction of systems characterized by different numbers of positive Lyapunov exponents is considered. To determine the lengths of characteristic phases of the system behavior the auxiliary system method has been used. The result of the work is the determination of the type of intermittency taking place near the boundary of generalized synchronization. In this case by calculation the statistical characteristics of the laminar phase lengths (distributions of the laminar phase lengths and the dependencies of the mean lengths of the laminar phases on the criticality parameters) it has been found that near the boundary of the synchronous regime the on-off intermittency is observed. It has been shown that the intermittent generalized synchronization in time-delayed systems is characterized by multistability. For these purposes a time-averaged measure of multistability depending on the value of the coupling parameter between systems has been calculated and compared with the behavior of the spectrum of Lyapunov exponents. It has been found that the multistability measure can be used to detect the generalized synchronization in time-delayed systems.

Synchronization of time-delay systems with impulsive delay via an average impulsive estimation approach.

We investigated synchronization of dynamic systems with mixed delays and delayed impulses. Using impulsive control method and the average impulsive interval approach, several Lyapunov sufficient conditions were given for ensuring synchronization in terms of impulsive perturbation and impulsive control, respectively. The derived conditions indicated that delays in continuous dynamical systems were flexible under impulsive perturbation and were not strictly dependent on the size of impulsive delays, and they may have a potential impact on synchronization of the considered system. In addition, applying the proposed concepts of average positive impulsive estimation and average impulsive estimation, we integrated the information in impulsive delay into the rate coefficient to eliminate the limitation of having the same threshold at each impulse point, while the impulsive delay maintained the synchronization effect. This was an improvement on the previous results obtained. Finally, we provided two numerical examples to illustrate the validity of our results.

Dual combination–combination synchronization of time delayed dynamical systems via adaptive sliding mode control under uncertainties and external disturbances

In this paper, the problem of dual combination–combination synchronization of time delay systems is studied for unknown parameters under uncertainties and external disturbances. By using adaptive sliding mode control method and Lyapunov stability criteria, nonlinear control functions and parameters updating equations are designed. This problem is first of its kind where synchronization of eight chaotic time delay systems is accomplished for unknown parameters. By performing rigorous calculations, sufficient condition is obtained for the asymptotically stable synchronization state. To implement the methodology two examples are presented. In first example, pairs of time delay Shimizu-Morioka and time delay Bhalekar–Gejji systems are considered while pairs of time delay advanced Lorenz and time delay Lorenz systems are considered in second example. Runge–Kutta method for delay-differential equations is used to perform numerical simulations. Theoretical and graphical treatments are presented to show the efficiency of the designed controllers and parameters updating equations.

Multi-switching Dual Combination Synchronization of Time Delay Dynamical Systems for Fully Unknown Parameters via Adaptive Control

In this paper, the problem of multi-switching dual combination synchronization of time delay systems is studied for fully unknown parameters. The controlling laws and parameters updating equations are designed by using adaptive control method and Lyapunov functional. In order to illustrate the problem, an example of time delay Rossler, Chen, Shimizu–Morioka, modified Lorenz, advanced Lorenz and Bhalekar–Gejji systems is discussed where time delay Rossler, Chen, Shimizu–Morioka, modified Lorenz systems are considered as leader systems and time delay advanced Lorenz and Bhalekar–Gejji systems are considered as follower systems. Theoretical analysis and graphical results are presented to depict the efficiency of the designed controllers and parameters updating equations. Graphical and analytical results are in excellent agreement.

Hyperchaos synchronization using univariate impulse control.

Rössler and Chen systems with time delay are shown to be hyperchaotic, which exhibits a more complex dynamics, including multiple positive Lyapunov exponents and infinite dimension. The hyperchaos has better application potential where hyperchaos synchronization is concerned. Univariate impulse control requires smaller perturbation to the response system, thus promising better performance. However, synchronization of two hyperchaotic systems using this control method is a challenging task due to the difficulty to guarantee synchronization stability using a minimum number of manipulated variables. In this paper, a univariate impulse control method is proposed for the synchronization of two hyperchaotic dynamics generated by time delay. A theorem is developed and proved to provide the sufficient conditions for the synchronization of time delay systems using the univariate impulse control. The upper bound of the impulse interval is proved to guarantee the asymptotic synchronization. Simulation and circuit experiment show the correctness of the analysis and the feasibility of the proposed method.

Synchronization of time delay systems with non-diagonal complex scaling functions

This paper deals with a kind of synchronization of dynamical systems with complex variables which is called generalized complex modified hybrid function projective synchronization (GCMHFPS) of time delay complex chaotic (hyperchaotic) systems. In other words, that the time delay complex systems can be synchronized up to a complex function transformation matrix. Moreover, the elements of the transformation matrix are complex functions of the states of drive system and time, where this matrix is not square. The idea of an active control method based on complex Krasovskii–Lyapunov functional is used to achieve GCMHFPS of time delay complex systems. Furthermore, based on the non-diagonal complex function transformation matrix, the modules and phases errors between one state of the complex response system and more than one of the states of the drive system are studied which have not been discussed before as far as we know. The analytical expression regarding the stability of this technique is derived and excellent agreement is found upon comparison with numerical calculations. In particular, we show through studying the time evolution of error, modulus and phase that the proposed scheme is effective for controlling time delay complex systems.

Controlled Synchronization of Time-Delayed System

The work focuses on synchronization of time-delay chaos systems. A theoretical analysis for time-delayed system is provided. By using the Lyapunov-Krasovskii approach, a synchronization condition is obtained. Finally, conditions are used for synchronization of time-delay chaos systems, and simulations verify the validity of obtained result.

Effect of noise on chaos synchronization in time-delayed systems: Numerical and experimental observations

We discuss the constructive role of noise (white and colored) in chaos synchronization in time-delayed systems. We first numerically investigate noise-induced synchronization (NIS) between two identical uncoupled Ikeda and Mackey–Glass systems. We find that synchronization occurs above a critical noise intensity that differs for different colors of noise. Synchronization onset is characterized by the value of the maximum transverse Lyapunov exponent. We then discuss the enhancement of chaos synchronization between two time-delayed systems when they are coupled unidirectionally. The effect of parameter mismatch for NIS is described in detail. We provide experimental evidence of NIS for a Mackey–Glass-like system in an electronic circuit using different colors of noise. An integration scheme for time-delayed systems in the presence of additive white and colored noise is discussed.

Comment on “Generalized projective synchronization in time-delayed systems: Nonlinear observer approach” [Chaos 19, 013102 (2009); 20, 029902 (2010)

In Chaos 19, 013102 (2009), the author proposed generalized projective synchronization for time delay systems using nonlinear observer and obtained sufficient condition to ensure projective synchronization for modulated time varying delay. There are concerns with the obtained conditions as the result was applicable only to trivial case of time varying delay τ[over dot](1)(t)=dτ(1)(t)/dt<1. In this paper, we note the drawbacks of the proposed sufficient condition. The new improved sufficient condition for ensuring the projective synchronization of time varying delayed systems is presented. The proposed new criteria have been verified by adopting the Ikeda system.

Design of coupling for synchronization in time-delayed systems

We report a design of delay coupling for targeting desired synchronization in delay dynamical systems. We target synchronization, antisynchronization, lag-and antilag-synchronization, amplitude death (or oscillation death), and generalized synchronization in mismatched oscillators. A scaling of the size of an attractor is made possible in different synchronization regimes. We realize a type of mixed synchronization where synchronization and antisynchronization coexist in different pairs of state variables of the coupled system. We establish the stability condition of synchronization using the Krasovskii-Lyapunov function theory and the Hurwitz matrix criterion. We present numerical examples using the Mackey-Glass system and a delay Rössler system.

Noise-enhanced phase synchronization in time-delayed systems

We investigate the phenomenon of noise-enhanced phase synchronization (PS) in coupled time-delay systems, which usually exhibit non-phase-coherent attractors with complex topological properties. As a delay system is essentially an infinite dimensional in nature with multiple characteristic time scales, it is interesting and crucial to understand the interplay of noise and the time scales in achieving PS. In unidirectionally coupled systems, the response system adjust all its time scales to that of the drive, whereas both subsystems adjust their rhythms to a single (main time scale of the uncoupled system) time scale in bidirectionally coupled systems. We find similar effects for both a common and an independent additive Gaussian noise.

Characteristics and synchronization of time-delay systems driven by a common noise

We investigate the characteristics of time-delay systems in the presence of Gaussian noise. We show that the delay time embedded in the time series of time-delay system with constant delay cannot be estimated in the presence noise for appropriate values of noise intensity thereby forbidding any possibility of phase space reconstruction. We also demonstrate the existence of complete synchronization between two independent identical time-delay systems driven by a common noise without explicitly establishing any external coupling between them.

Erratum: “Generalized projective synchronization in time-delayed systems: Nonlinear observer approach” [Chaos 19, 013102 (2009)

Erratum: “Generalized projective synchronization in time-delayed systems: Nonlinear observer approach” [Chaos 19, 013102 (2009)

Nonlinear active observer-based generalized synchronization in time-delayed systems

When two different chaotic oscillators are coupled, generalized synchronization can occur. It may imply a very complicated relation between the states of drive and response systems. We propose a method that can be used to detect and characterize the generalized synchronization in modulated time-delayed systems. Using Krasovskii–Lyapunov theory, sufficient condition for generalized synchronization is derived. The proposed technique has been applied to synchronize prototype and Ikeda models by numerical simulation.

Synchronization of time-delayed systems with chaotic modulation and cryptography

This paper presents a method of synchronization between two time-delayed systems where the delay times are modulated by a common chaotic signal of the driving system. The technique is well applied to two identical autonomous continuous-time-delayed systems with numerical simulations. Finally, a new method of encryption is generated for digital messages. This method is illustrated with two different encryption processes for text as well as picture messages.

Generalized projective synchronization in time-delayed systems: Nonlinear observer approach

In this paper, we consider the projective-anticipating, projective, and projective-lag synchronization in a unified coupled time-delay system via nonlinear observer design. A new sufficient condition for generalized projective synchronization is derived analytically with the help of Krasovskii-Lyapunov theory for constant and variable time-delay systems. The analytical treatment can give stable synchronization (anticipatory and lag) for a large class of time-delayed systems in which the response system's trajectory is forced to have an amplitude proportional to the drive system. The constant of proportionality is determined by the control law, not by the initial conditions. The proposed technique has been applied to synchronize Ikeda and prototype models by numerical simulation.

Conditions for common-noise-induced synchronization in time-delay systems

We studied synchronization induced by a common external noise in scalar time-delay systems. We have found a set of sufficient conditions for the synchronization. This set of conditions shows that the synchronization occurs in a wide class of time-delay systems. Numerical evidence for the analytically obtained conditions is also presented.

Transition from phase to generalized synchronization in time-delay systems

The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piecewise linear and in coupled Mackey–Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase-coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence-based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence, joint probability of recurrence, and similarity of probability of recurrence. We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.

Realization of synchronization in time-delayed systems with stochastic perturbation

Since noise is ubiquitous in both nature and artificial systems, the stochastic perturbation influence on the dynamics of the unidirectionally coupled Ikeda models is investigated in this paper. On the one hand, sufficient conditions on the complete synchronization between these noise-perturbed and chaotic models are mathematically established, and an estimation of the sample transverse Lyapunov exponent is rigorously derived. On the other hand, specific examples and their numerical simulations are provided to illustrate the feasibility of our theoretical results. Moreover, the results on the Ikeda models are further generalized to a wide class of coupled nonlinear systems with multiple time delays and a common additive noise. It is believed that the idea and approach developed in this paper could be further generalized to investigate some other problems on chaos synchronization and chaos control with stochastic perturbation.

Erratum: Synchronization of time-delayed systems [Phys. Rev. E76, 036212 (2007)

Received 14 November 2007DOI:https://doi.org/10.1103/PhysRevE.76.059905©2007 American Physical Society