Linear observer based projective synchronization in delay Rössler system

A new type of linear observer based projective, projective anticipating and projective lag synchronization of time-delayed Rössler system is studied. Along with this, the approach arbitrarily scales a drive system attractor and hence a similar chaotic attractor of any desired scale can be realized with the help of a synchronizing scaling factor. A scalar synchronizing output is considered where the output equation includes both the delay and non-delay terms of the nonlinear function. The condition for synchronization is derived analytically and the values of the coupling parameters are obtained. Analytical results are verified through numerical investigation and the effect of modulated time delay in the method is discussed. An important aspect of this method is that it does not require the computation of conditional Lyapunov exponents for the verification of synchronization.     

Various synchronization phenomena in bidirectionally coupled double scroll circuits

In this paper, the various cases of synchronization phenomena investigated in a system of two bidirectionally coupled double scroll circuits, were studied. Complete synchronization, inverse lag synchronization, and inverse π-lag synchronization are the observed synchronization phenomena, as the coupling factor is varied. The inverse lag synchronization phenomenon in mutually coupled identical oscillators is presented for the first time. As the coupling factor is increased, the system undergoes a transition from chaotic desynchronization to chaotic complete synchronization, while inverse lag synchronization and inverse π-lag synchronization are observed for greater values of the coupling factor, depending on the initial conditions of the state variables of the system. Inverse π-lag synchronization in coupled nonlinear oscillators is a special case of lag synchronization, which is also presented for the first time.     

Synchronization in multiple time delay chaotic laser diodes subject to incoherent optical feedbacks and incoherent optical injection

We present the first report of the synchronization regimes in both unidirectionally and bidirectionally coupled multiple time delay chaotic laser diodes subject to incoherent optical feedbacks and incoherent optical injection. We derive the existence conditions and numerically study the stability for lag, complete, and anticipating synchronization regimes and cascaded synchronization. We also study in detail the effect of parameter mismatches and noise on the synchronization quality. It is emphasized that sensitivity of the synchronization quality to parameter mismatches can lead to a high level of security due to the difficulty in replicating the receiver laser. We show that the injection current and feedback delay times are highly important parameters from this point of view.     

Synchronization transition in degenerate optical parametric oscillators induced by nonlinear coupling

In this paper, the synchronization of certain degenerate optical parametric oscillators is investigated in detail. Complete and/or partial synchronization can be reached when linear controller, constructed by the real part or imaginary part of the subharmonic mode, is imposed on the chaotic degenerate optical parametric oscillators with appropriate coupling intensity. The Lyapunov exponents under different coupling coefficients are calculated to find the critical condition for complete synchronization. Transition from complete synchronization to partial synchronization is observed when nonlinear coupling is introduced into the two chaotic oscillators. It is found that synchronization of chaotic oscillators keeps robust when the intensity of the nonlinear coupling is less than the intensity of the linear coupling; the complete synchronization state is destructed and transient period for partial synchronization is in certain delay when the intensity of the nonlinear coupling is beyond the intensity of the linear coupling.     

Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems

In this paper, a new type of anticipating synchronization, called time-varying anticipating synchronization, is defined firstly. Then novel adaptive schemes for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems are designed based on the Lyapunov function and invariance principle. The update gain of coupling strength can be automatically adapted to a suitable strength depending on the initial values and can be properly chosen to adjust the speed of achieving synchronization, so these schemes are analytical and simple to implement in practice. A classical chaotic dynamical system is used to demonstrate the effectiveness of the proposed adaptive schemes with or without parameter uncertainties.     

On synchronization of a forced delay dynamical system via the Galerkin approximation

A forced scalar delay dynamical system is analyzed from the perspective of bifurcation and synchronization. In general first order differential equations do not exhibit chaos, but introduction of a delay feedback makes the system infinite dimensional and shows chaoticity. In order to study the dynamics of such a system, Galerkin projection technique is used to obtain a finite dimensional set of ordinary differential equations from the delay differential equation. We compare the results of simulation with those obtained from direct numerical simulation of the delay equation to ascertain the accuracy of the truncation process in the Galerkin approximation. We have considered two cases, one with five and the other with eight shape functions. Next we study two types of synchronization by considering coupling of the above derived equations with a forced dynamical system without delay. Our analysis shows that it is possible to have synchronization between two such systems. It has been shown that the chaotic system with delay feedback can drive the system without delay to achieve synchronization and the opposite case is also equally valid. This is confirmed by the evaluation of the conditional Lyapunov exponents of the systems.     

Lag synchronizing chaotic system based on a single controller

Lag synchronization of chaotic system is investigated. Three kinds of schemes are proposed to lag synchronize Chen chaotic system. All the three schemes need only a single controller to realize lag synchronization. Especially in the last two schemes, only one state variable is contained in controller, which is of important significance on using chaos lag synchronization for applications. Finally numerical simulations are provided to show the effectiveness of the developed methods.     

Chaotic synchronization of two mutually coupled semiconductor lasers for optoelectronic logic gates

A physical model of the fundamental configuration of two mutually coupled semiconductor lasers is presented for logic-gate applications, and the principles of optoelectronic logic computing based on chaotic synchronization or chaotic de-synchronization are defined. Two laser diodes were coupled via injection of each into the opposite laser and became chaotic; our analysis showed that the oscillation derives from chaotic fluctuations after a progression from stability to period-doubling by varying the coupling factor, delay time or detuning. Chaotic synchronization is achieved between the two lasers through the coupling, where we found chaotic and quasi-periodic synchronization regions. Based on the chaotic synchronization system, three optoelectronic logic gates can be implemented by modulating the laser diode current to synchronize or de-synchronize the two chaotic states. Finally, we studied the effects of resynchronization time on logic gate function in a practical implementation of the system. Numerical results show the validity and feasibility of the method.     

Synchronization in time-discrete model of two electrically coupled spike-bursting neurons

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.     

Robust adaptive neural network synchronization controller design for a class of time delay uncertain chaotic systems

In this paper, a robust adaptive neural network synchronization controller is proposed for two chaotic systems with input time delay and uncertainty. The studied chaotic system may possess a wide class of nonlinear time-delayed input uncertainty. The radial basis function (RBF) neural network is used to approximate the unknown continuous bounded function item of the time delay uncertainty via appropriate weight value updated law. With the output of RBF neural network, a robust adaptive synchronization control scheme is presented for the time delay uncertain chaotic system. Finally, a simulation example is used to illustrate the effectiveness of the proposed synchronization control scheme.     

Lag and anticipatory synchronization based parameter estimation scheme in modulated time-delayed systems

In this article, we have shown that using delay dynamical systems as base, one can use the modulation of the various parameters, to transmit multiple signals through a single chaotic time series. It is shown that under certain conditions the original signals and parameters of the driving signals can be recovered with the help of an adaptive demodulator. An important aspect of the present method is that the communication is possible between two time-delayed systems with parameter mismatch using lag and anticipatory synchronization. The whole idea is presented on the basis of Mackey–Glass system, with variable time delay.     

Observer-based lag synchronization between two different complex networks

In this paper, some new criteria for lag synchronization between two or more complex networks are proposed based on the theory of state observer. Some adaptive controllers are designed to make the drive and response systems achieve lag synchronization, no matter whether the nodes in the two systems are with the same dynamical character or the coupling configuration matrices are nonidentical. In addition, based on the output coupling, the amount of coupling variables between two connected nodes is flexible, which can save a lot of channel resources, simplify the network topology and has more significant meanings in engineering applications. At last, the effects of the lag synchronization criteria are verified through some simulation experiments.     

Adaptive generalized function projective lag synchronization of different chaotic systems with fully uncertain parameters

In this paper, a novel projective synchronization scheme called adaptive generalized function projective lag synchronization (AGFPLS) is proposed. In the AGFPLS method, the states of two different chaotic systems with fully uncertain parameters are asymptotically lag synchronized up to a desired scaling function matrix. By means of the Lyapunov stability theory, an adaptive controller with corresponding parameter update rule is designed for achieving AGFPLS between two diverse chaotic systems and estimating the unknown parameters. This technique is employed to realize AGFPLS between uncertain Lü chaotic system and uncertain Liu chaotic system, and between Chen hyperchaotic system and Lorenz hyperchaotic system with fully uncertain parameters, respectively. Furthermore, AGFPLS between two different uncertain chaotic systems can still be achieved effectively with the existence of noise perturbation. The corresponding numerical simulations are performed to demonstrate the validity and robustness of the presented synchronization method.     

Projective synchronization of time‒delayed chaotic systems with unknown parameters using adaptive control method

The present article aims to study the projective synchronization between two identical and non?identical time?delayed chaotic systems with fully unknown parameters. Here the asymptotical and global synchronization are achieved by means of adaptive control approach based on Lyapunov–Krasovskii functional theory. The proposed technique is successfully applied to investigate the projective synchronization for the pairs of time?delayed chaotic systems amongst advanced Lorenz system as drive system with multiple delay Rössler system and time?delayed Chua's oscillator as response system. An adaptive controller and parameter update laws for unknown parameters are designed so that the drive system is controlled to be the response system. Numerical simulation results, depicted graphically, are carried out using Runge–Kutta Method for delay?differential equations, showing that the design of controller and the adaptive parameter laws are very effective and reliable and can be applied for synchronization of time?delayed chaotic systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes is investigated in this paper. Based on Barbalat’s lemma, some sufficient synchronization criteria are derived by applying the nonlinear feedback control. Although previous work studied function projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In our work, the dynamics of the nodes of the complex networks are any chaotic systems without the limitation of the partial linearity. In addition, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. Numerical simulations further verify the effectiveness and feasibility of the proposed synchronization method. Numeric evidence shows that the synchronization rate is sensitively influenced by the feedback strength, the time delay, the network size and the network topological structure.     

Tracking the state of the delay hyperchaotic Lü system using the coullet chaotic system via a single controller

In this article, a partial synchronization scheme is proposed based on Lyapunov stability theory to track the signal of the delay hyperchaotic Lü system using the Coullet system based on only one single controller. The proposed tracking control design has two advantages: only one controller is adopted in our approach and it can allow us to drive the hyperchaotic system to a simple chaotic system even with uncertain parameters. Numerical simulation results are given to demonstrate the effectiveness and robustness of the proposed partial synchronization scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 125–130, 2016     

Synchronization analysis of complex‐Variable chaotic systems with discontinuous unidirectional coupling

This article is concerned with the problem of synchronization between two uncertain complex‐variable chaotic systems with parameters perturbation and discontinuous unidirectional coupling. Based on the stability theory and comparison theorem of differential equations, some sufficient conditions for the complete synchronization and generalized synchronization are obtained. The theoretical results show that the two uncertain complex‐variable chaotic systems with discontinuous unidirectional coupling can achieve synchronization if the time‐average coupling strength is large enough. Finally, numerical examples are examined to illustrate the feasibility and effectiveness of the analytical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 343–355, 2016     

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.     

Lag synchronization of hyperchaos with application to secure communications

In this paper, hyperchaotic lag synchronization is restated as a nonlinear and lag-in-time observer design issue. This approach leads to a systematic tool, which guarantees the lag synchronization of a wide class of chaotic or hyperchaotic systems via a scalar signal. By exploiting this result, we propose a hyperchaos-based cryptosystem scheme that combines the conventional cryptographic methods and the lag synchronization of chaotic circuits. The computer simulation results show that the lag synchronization scheme and the cryptosystem proposed in this paper are both feasible.     

Projective synchronization with different scale factors in a driven–response complex network and its application in image encryption

We study projective (anticipatory, exact and lag) synchronization of time-delayed chaotic systems in a driven–response complex network (DRCN), where the nodes are not partially linear and the scale factors are different from each other. Two synchronization criteria are presented; the first is easy to satisfy with simple limitation and the second relaxes the previous limitation and can be executed only by adding a simple linear feedback control to every node. As a special case, projective synchronization in a DRCN without time delay is analyzed by using a different method. The corresponding synchronization criterion is given, which can be realized by adopting single-node adaptive control. This paper extends previous work, where the nodes of the DRCN are partially linear, there is limitation of the coupling matrix or the scale factors of the nodes are all equal. Numerical examples are given to demonstrate the effectiveness of the proposed synchronization criteria. Due to the complexity of the different scale factors, we select the chaotic data from the DRCN presented in this paper and apply the data selected in image encryption. Simulation results show that our method is efficient with a large key space, high sensitivity to initial conditions and high speed.