Different Types of Synchronization in Time-Delayed Systems

We investigate different types of synchronization between two unidirectionally nonlinearly coupled identical delay- differential systems related to optical bistable or hybrid optical bistable devices. This system can represent some kinds of delay-differential models, i.e. Ikeda model, Vall~e model, sine-square model, Mackey Glass model, and so on. We find existence and sufficient stability conditions by theoretical analysis and test the correctness by＂ numerical simulations. Lag, complete and anticipating synchronization are observed, respectively. It is found that the time-delay system can be divided into two parts~ one is the instant term and the other is the delay term. Synchronization between two identical chaotic systems can be derived by adding a coupled term to the delay term in the driven system.     

Average Synchronization and Temporal Order in a Noisy Neuronal Network with Coupling Delay

Average synchronization and temporal order characterized by the rate of firing are studied in a spatially extended network system with the coupling time delay, which is locally modelled by a two-dimensional Rulkov map neuron. It is shown that there exists an optimal noise level, where average synchronization and temporal order are maximum irrespective of the coupling time delay. Furthermore, it is found that temporal order is weakened when the coupling time delay appears. However, the coupling time delay has a twofold effect on average synchronization, one associated with its increase, the other with its decrease. This clearly manifests that random perturbations and time delay play a complementary role in synchronization and temporal order.     

Directed transient long-range transport in a slowly driven Hamiltonian system of interacting particles

We study the Hamiltonian dynamics of a one-dimensional chain of linearly coupled particles in a spatially periodic potential which is subjected to a time-periodic mono-frequency external field. The average over time and space of the related force vanishes and hence, the system is effectively without bias which excludes any ratchet effect. We pay special attention to the escape of the entire chain when initially all of its units are distributed in a potential well. Moreover for an escaping chain we explore the possibility of the successive generation of a directed flow based on large accelerations. We find that for adiabatic slope-modulations due to the ac-field transient long-range transport dynamics arises whose direction is governed by the initial phase of the modulation. Most strikingly, that for the driven many particle Hamiltonian system directed collective motion is observed provides evidence for the existence of families of transporting invariant tori confining orbits in ballistic channels in the high-dimensional phase spaces.     

Projective cluster synchronization in drive-response dynamical networks

In this paper, we study the projective cluster synchronization in a drive-response dynamical network with 1+N coupled partially linear chaotic systems. Because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable, pinning control ideas are adopted to direct the different scaling factors onto the desired values. It is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.     

Anomalous mobility and current reversals in inertial deterministic ratchets

We analyze the transport properties of inertial deterministic rocking ratchets in the presence of an external constant force. For small values of this load, we can obtain a positive current for a negative load, and vice versa. This phenomenon, in which the direction of the current is opposed to the sign of the external force, is a signature of anomalous negative mobility. We show that this anomalous mobility is possible in the deterministic case, and explain this phenomenon as current reversals associated to bifurcations in an inertial deterministic rocking ratchet in the presence of an external load.     

Synchronization and control of chaos in coupled chaotic multimode Nd:YAG lasers

In this Letter we numerically investigate the dynamics of a system of two coupled chaotic multimode Nd:YAG lasers with two mode and three mode outputs. Unidirectional and bidirectional coupling schemes are adopted; intensity time series plots, phase space plots and synchronization plots are used for studying the dynamics. Quality of synchronization is measured using correlation index plots. It is found that for laser with two mode output bidirectional direct coupling scheme is found to be effective in achieving complete synchronization, control of chaos and amplification in output intensity. For laser with three mode output, bidirectional difference coupling scheme gives much better chaotic synchronization as compared to unidirectional difference coupling but at the cost of higher coupling strength. We also conclude that the coupling scheme and system properties play an important role in determining the type of synchronization exhibited by the system.     

Investigation of a Unified Chaotic System and Its Synchronization by Simulations*

We investigate a unified chaotic system and its synchronization including feedback synchronization and adaptive synchronization by numerical simulations. We propose a new dynamical quantity denoted by K, which connects adaptive synchronization and feedback synchronization, to analyze synchronization schemes. We find that K can estimate the smallest coupling strength for a unified chaotic system whether it is complete feedback or one-sided feedback. Based on the previous work, we also give a new dynamical method to compute the leading Lyapunov exponent.     

Rhythm Synchronization of Coupled Neurons with Temporal Coding Scheme

Encoding information by firing patterns is one of the basic neural functions, and synchronization is important collective behaviour of a group of coupled neurons. Taking account of two schemes for encoding information （that is, rate coding and temporal coding）, rhythm synchronization of coupled neurons is studied. There are two types of rhythm synchronization of neurons： spike and burst synchronizations. Firstly, it is shown that the spike synchronization is equivalent to the phase synchronization for coupled neurons. Secondly, the similarity function of the slow variables of neurons, which have relevant to the bursting process, is proposed to judge the burst synchronization. It is also found that the burst synchronization can be achieved more easily than the spike synchronization, whatever the firing patterns of the neurons are. Hence the temporal encoding scheme, which is closely related to both the spike and burst synchronizations, is more comprehensive than the rate coding scheme in essence.     

The impulsive control synchronization of the drive-response complex system

This Letter investigates projective synchronization between the drive system and response complex dynamical system. An impulsive control scheme is adapted to synchronize the drive-response dynamical system to a desired scalar factor. By using the stability theory of the impulsive differential equation, the criteria for the projective synchronization are derived. The feasibility of the impulsive control of the projective synchronization is demonstrated in the drive-response dynamical system.     

Global synchronization in arrays of coupled networks with one single time-varying delay coupling

Global synchronization in arrays of coupled networks with one single time-varying delay coupling is investigated in this Letter. A general linear coupled network with a time-varying coupling delay is proposed and its global synchronization is further discussed. Some sufficient criteria are derived based on Lyapunov functional and linear matrix inequality (LMI). It is shown that under one single delay coupling, the synchronized state changes, which is different from the conventional synchronized solution. Moreover, the degree of the nodes and the inner delayed coupling matrix play key roles in the synchronized state. In particular, the derivative of the time-varying delay can be any given value. Finally, numerical simulations are given to illustrate the theoretical results.     

Detecting synchronization in coupled stochastic ecosystem networks

Instantaneous phase difference, synchronization index and mutual information are considered in order to detect phase transitions, collective behaviours and synchronization phenomena that emerge for different levels of diffusive and reactive activity in stochastic networks. The network under investigation is a spatial 2D lattice which serves as a substrate for Lotka-Volterra dynamics with 3rd order nonlinearities. Kinetic Monte Carlo simulations demonstrate that the system spontaneously organizes into a number of asynchronous local oscillators, when only nearest neighbour interactions are considered. In contrast, the oscillators can be correlated, phase synchronized and completely synchronized when introducing different interactivity rules (diffusive or reactive) for nearby and distant species. The quantitative measures of synchronization show that long distance diffusion coupling induces phase synchronization after a well defined transition point, while long distance reaction coupling induces smeared phase synchronization.     

Phase synchronization in tilted deterministic ratchets

We study phase synchronization for a ratchet system. We consider the deterministic dynamics of a particle in a tilted ratchet potential with an external periodic forcing, in the overdamped case. The ratchet potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of external periodic forcing. This oscillator has an intrinsic frequency that can be entrained with the frequency of the external driving. We introduced a linear phase through a set of discrete time events and the associated average frequency, and show that this frequency can be synchronized with the frequency of the external driving. In this way, we can properly characterize the phenomenon of synchronization through Arnold tongues, which represent regions of synchronization in parameter space, and discuss their implications for transport in ratchets.     

A data-analysis method for decomposing synchronization variability of anticipatory systems into stochastic and deterministic components

Synchronization has been shown to be a valuable concept in the field of nonlinear dynamics and dynamical systems in general. Deviation from perfect synchronization results from an interplay of deterministic coupling forces and stochastic fluctuating forces. When the exact details of these two sources of variance are unknown, it becomes useful to estimate them directly from data. To this end, we develop a data analysis method for estimating parameters associated with these deterministic and stochastic components. The method relies on separating their respective contributions to synchronization error. We focus on the case where a slave system synchronizes with the future of a master system, so-called anticipating synchronization.     

A novel study of parity and attractor in the time reversed Lorentz system

In this Letter, a new effective approach to achieve adaptive synchronization is proposed. Via using Ge-Yao-Chen (GYC) partial region stability theory and pragmatical asymptotically stability theorem, the numerical simulation results show that the states errors and parameter errors approach to zero much more exactly and efficiently than traditional method. The time reversed Lorenz system (called historical Lorenz system in this Letter) is introduced and used for example in this Letter. The simulation results are given in figures and tables for comparison between the new approach and traditional one to show the effectiveness and feasibility of our new strategy.     

Preserving synchronization under matrix product modifications

In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under modification through matrix multiplication. The objective is to show that under a defined multiplicative group, hyperbolic critical points are preserved along the stable and unstable manifolds. The properties of this multiplicative group were determined through the use of simultaneous Jordan decomposition. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of different modified systems.     

Synchronization of chaos in unidirectionally and bidirectionally coupled multiple time delay laser diodes with electro-optical feedback

We report on chaos synchronization in both unidirectionally and bidirectionally coupled multiple time delay laser diodes with electro-optical feedback. We derive existence and sufficient stability conditions for the synchronization regimes. We calculate the Lyapunov exponents, information dimension, and Kolmogorov-Sinai entropy for a single and double delay time lasers to demonstrate that multiple time delay laser system can offer higher complexity than a single time delay laser. We demonstrate that in coupled multiple time delay lasers additional feedback(s) can play a stabilizing role. We compare the synchronization quality for closed loop and open loop receiver laser configurations and find better synchronization quality for partially open loop receiver (when the receiver laser has only one feedback loop), than the open loop receiver configuration (when the receiver contains no feedback loops). We also study the effect of the feedback phase on the correlation coefficient between the interacting laser systems. Analytical results are fully supported by numerical simulations.     

The effect of time-delay on anomalous phase synchronization

Anomalous phase synchronization in nonidentical interacting oscillators is manifest as the increase of frequency disorder prior to synchronization. We show that this effect can be enhanced when a time-delay is included in the coupling. In systems of limit-cycle and chaotic oscillators we find that the regions of phase disorder and phase synchronization can be interwoven in the parameter space such that as a function of coupling or time-delay the system shows transitions from phase ordering to disorder and back.     

Burst synchronization of electrically and chemically coupled map-based neurons

Burst synchronization and burst dynamics of a system consisting of two map-based neurons coupled through electrical or chemical synapses are discussed. Some basic characteristic quantities are introduced to describe burst synchronization and burst dynamics of neurons. It is observed that excitatory coupling leads to in-phase burst synchronization but inhibitory coupling results in anti-phase one. By using the basic characteristics of burst dynamics, the effects of the intrinsic bursting properties and the coupling schemes on complex bursting behaviors are also presented for both inhibitory and excitatory couplings. The results are instructive to identify bursting behaviors through experimental data.     

Synchronization Scheme for Uncertain Chaotic Systems via RBF Neural Network

A sliding mode adaptive synchronization controller is presented with a neural network of radial basis function （RBF） for two chaotic systems. The uncertainty of the synchronization error system is approximated by the RBF neural network. The synchronization controller is given based on the output of the RBF neural network. The proposed controller can make the synchronization error convergent to zero in 5s and can overcome disruption of the uncertainty of the system and the exterior disturbance. Finally, an example is given to illustrate the effectiveness of the proposed synchronization control method.     

Synchronization of mutually coupled systems

General operation conditions to obtain anticipated or retarded synchronization in a mutually coupled system with feedback are analyzed. Different from a unidirectionally coupled system, which type of synchronization will occur in a mutually coupled system does not solely depend on the difference between the feedback-delay time and the coupling-delay time. We find that in addition to the feedback/coupling delay times, feedback strengths and coupling strengths all determine whether the system will preform anticipated or retarded synchronization. Experimental implementation using semiconductor lasers subject to optoelectronic feedback is provided as an example. A set of experimental data is presented and discussed.