Synchronization of time-delay chaotic systems on small-world networks with delayed coupling

By using the well-known Ikeda model as the node dynamics, this paper studies synchronization of time-delay systems on small-world networks where the connections between units involve time delays. It shows that, in contrast with the undelayed case, networks with delays can actually synchronize more easily. Specifically, for randomly distributed delays, time-delayed mutual coupling suppresses the chaotic behaviour by stabilizing a fixed point that is unstable for the uncoupled dynamical system.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Synchronization transition of identical phase oscillators in a directed small-world network

We numerically study a directed small-world network consisting of attractively coupled, identical phase oscillators. While complete synchronization is always stable, it is not always reachable from random initial conditions. Depending on the shortcut density and on the asymmetry of the phase coupling function, there exists a regime of persistent chaotic dynamics. By increasing the density of shortcuts or decreasing the asymmetry of the phase coupling function, we observe a discontinuous transition in the ability of the system to synchronize. Using a control technique, we identify the bifurcation scenario of the order parameter. We also discuss the relation between dynamics and topology and remark on the similarity of the synchronization transition to directed percolation.     

Synchronization of relaxational self-oscillating systems: Synchronization in neural networks

A general problem of the synchronization and mutual synchronization of relaxational self-oscillating systems is formulated. A direct way of describing the synchronization of relaxational systems on the basis of Kronecker’s inequalities is proposed. The solution to the problem formulated by N. Wiener and A. Rosenbluth of forming a single rhythm in a system of coupled relaxational oscillators is described. Specific transient processes in the synchronization of relaxational systems are considered. Burst synchronization in neural networks and synchronization in distributed relaxational systems are also described.     

Synchronization of slow-fast systems

We provide an overview over the following eleven contributions on superconductivity in copper-oxygen and iron-based compounds. The main objective of this volume is an improved general understanding of superconductivity at high transition temperatures. The key questions on the way towards understanding superconducting pairing beyond electron-phonon coupling are spelled out, and the present status of theoretical reasoning is summarized. The crucial experiments, their results and interrelations are discussed. The central result is that fluctuations of spin and charge contribute substantially to superconductivity and also to other ordering phenomena. Methodically, the simultaneous analysis of results obtained from different experimental techniques such as photoelectron spectroscopy and neutron scattering, on one and the same sample, turned out to be of pivotal importance.     

Synchronization and emergence in complex systems

We show how novel behaviour can emerge in complex systems at the global level through synchronization of the activities of their constituent units. Two mechanisms are suggested for the emergence, namely non-diffusive coupling and time delays. In this way, simple units can synchronize to display complex dynamics, or conversely, simple dynamics may arise from complex constituents.     

Synchronization in systems with bimodal dynamics

Considering a prototypic model of a bimodal oscillator we investigate the synchronization of the internal time scales for a system with interacting fast and slow oscillatory modes. Particular emphasis is given to the transition between mode-locked and mode-unlocked chaos. It is shown that this transition involves a homoclinic bifurcation in which the synchronized chaotic attractor loses its band structure. For two coupled bimodal oscillators we illustrate the presence of separate synchronization regions for the fast and the slow modes. The dependence of these regions on the mismatch and coupling parameters is studied.     

Synchronization in coupled Ikeda delay systems

In this work, we demonstrate the use of a Field Programmable Gate Array (FPGA) as a physical platform for realizing chaotic delay differential equations (DDE). Moreover, using our platform, we also experimentally study the synchronization between two time delayed systems. We illustrate two different experimental approaches – one is hardware co-simulation (using a Digilent Atlys with a Xilinx Spartan-6 FPGA) and the other is analog output (using a Terasic DE2-115 with an Altera Cyclone IV E FPGA).     

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.     

耦合非线性振子系统的同步研究

研究了考虑振子振幅效应的耦合极限环系统的同步.研究表明，耦合极限环系统的序参量随耦合强度的增加呈现非单调变化，并且出现若干不可微的点；平均频率随耦合强度的变化过程表现为同步分岔树结构；在临界点处出现了相速度的滑移、锁定和相速度差的开关阵发现象，开关阵发的平均周期具有很好的标度关系；振子的平均振幅随相同步的进程实际上是由均匀化逐渐分岔而达到非均匀化的过程，振子振幅的变化范围在临界点处突然减小. 关键词： 耦合极限环系统 同步 振幅效应     

Synchronization of reconstructed dynamical systems

The problem of constructing synchronizing systems to observed signals is approached from a data driven perspective, in which it is assumed that neither the drive nor the response systems are known explicitly but have to be derived from the observations. The response systems are modeled by utilizing standard methods of nonlinear time series analysis applied to sections of the driving signals. As a result, synchronization is more robust than what might be expected, given that the reconstructed systems are only approximations of the unknown true systems. Successful synchronization also may be accomplished in cases where the driving signals result from nonlinearly transformed chaotic states. The method is readily extended and applied to limited real-time predictions of chaotic signals.     

Synchronization in Nonlinear Three-Dimensional Double-Diffusive Convection systems

The twelve-mode truncation of a three-dimensional double-diffusive convection model is studied.This system possesses an interchange symmetry (will be defined in the text). Exact and rough synchronizations based on this symmetry are observed.     

Synchronization in networks of spatially extended systems

Synchronization processes in networks of spatially extended dynamical systems are analytically and numerically studied. We focus on the relevant case of networks whose elements (or nodes) are spatially extended dynamical systems, with the nodes being connected with each other by scalar signals. The stability of the synchronous spatio-temporal state for a generic network is analytically assessed by means of an extension of the master stability function approach. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. The efficiency and reliability of this method is illustrated numerically with networks of beam-plasma chaotic systems (Pierce diodes). We discuss also how the revealed regularities are expected to take place in other relevant physical and biological circumstances.     

Synchronization in multicell systems exhibiting dynamic plasticity

Collective behaviour in multicell systems arises from exchange of chemicals/ signals between cells and may be different from their intrinsic behaviour. These chemicals are products of regulated networks of biochemical pathways that underlie cellular functions, and can exhibit a variety of dynamics arising from the non-linearity of the reaction processes. We have addressed the emergent synchronization properties of a ring of cells, diffusively coupled by the end product of an intracellular model biochemical pathway exhibiting non-robust birhythmic behaviour. The aim is to examine the role of intercellular interaction in stabilizing the non-robust dynamics in the emergent collective behaviour in the ring of cells. We show that, irrespective of the inherent frequencies of individual cells, depending on the coupling strength, the collective behaviour does synchronize to only one type of oscillations above a threshold number of cells. Using two perturbation analyses, we also show that this emergent synchronized dynamical state is fairly robust under external perturbations. Thus, the inherent plasticity in the oscillatory phenotypes in these model cells may get suppressed to exhibit collective dynamics of a single type in a multicell system, but environmental influences can sometimes expose this underlying plasticity in its collective dynamics.      

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

对称非线性耦合混沌系统的同步

研究两个对称非线性耦合混沌系统的同步问题.通过对系统线性项与非线性项的适当分离， 构造一个特殊的非线性耦合项，发现在耦合强度α＝05附近的某一区域里存在稳定的 混沌同步现象.提供判断同步误差稳定性的方程，利用线性系统的稳定性分析准则和条件Lya punov指数来检验同步状态的稳定性.新方法适用于连续时间系统的混沌同步，也适用于具有 两个（或多于两个）正Lyapunov指数的超混沌系统.以Lorenz系统，超混沌Rssler 系统作 为算例，数值模拟结果证实所提新方法的有效性. 关键词： 混沌 同步 非线性耦合 稳定性准则 超混沌     

Synchronization rate of synchronized coupled systems

Synchronization phenomena, an emergent property in networks of interacting dynamical elements, are widely observed in nature, and have become the subject of intense research. Here we will investigate the synchronization rate problem in coupled limit-cycle and chaotic oscillators. Based on the mode decomposition method and Gershgörin's discs theorem, some sufficient conditions for synchronization of coupled systems are obtained, and a synchronization rate is then derived. Such a synchronization rate indicates that the error functions between state variables of underlying individual systems tend to zero in the exponential form as time tends to the infinity. Several numerical examples are also given to validate the theoretical results.     

Overview: Synchronization and patterns in complex systems

The theory of complex systems, such as neural assemblies or lattices of chaotic oscillators has generated many new problems including the synchronization or regularization of the cooperative behavior of systems consisting of chaotic elements, regular spatial patterns in "chaotic" lattices, and so on. A number of these problems were discussed at the International School in Nonlinear Science-95 (Nizhniy Novgorod, Russia). In this overview we try to formulate some of the most interesting problems that were discussed at that meeting. (c) 1996 American Institute of Physics.     

Synchronization of spectral components in unidirectionally coupled systems

The object of investigation is synchronization of spectral components in unidirectionally coupled chaotic systems. It is shown that the behavior of the spectral components in this case is governed by the amount of detuning between interacting systems. Universal laws of spectral component synchronization are discovered.