Synchronization of oscillators in complex networks

Theory of identical or complete synchronization of identical oscillators in arbitrary networks is introduced. In addition, several graph theory concepts and results that augment the synchronization theory and a tie in closely to random, semirandom, and regular networks are introduced. Combined theories are used to explore and compare three types of semirandom networks for their efficacy in synchronizing oscillators. It is shown that the simplest k-cycle augmented by a few random edges or links are the most efficient network that will guarantee good synchronization.      

Collective behavior of electronic fireflies

A simple system composed of electronic oscillators capable of emitting and detecting light-pulses is studied. The oscillators are biologically inspired, there are designed for keeping a desired light intensity, W, in the system. From another perspective, the system behaves like modified integrate and fire type neurons that are pulse-coupled with inhibitory type interactions: the firing of one oscillator delays the firing of all the others. Experimental and computational studies reveal that although no direct driving force favoring synchronization is considered, for a given interval of W phase-locking appears. This weak synchronization is sometimes accompanied by complex dynamical patterns in the flashing sequence of the oscillators.     

Synchronization of indirectly coupled Lorenz oscillators: An experimental study

The dynamics of indirectly coupled Lorenz circuits is investigated experimentally. The in-phase and anti-phase synchronization of indirectly coupled chaotic oscillators reported in Phys. Rev. E 81, 046216 (2010) is verified by physical experiments with electronic circuits. Two chaotic systems coupled through a common dynamic environment shows the verity of synchronization behaviours such as anti-phase synchronization, in-phase synchronization, identical synchronization, anti-synchronization, etc.     

Synchronization and basin bifurcations in mutually coupled oscillators

Synchronization behaviour of two mutually coupled double-well Duffing oscillators exhibiting cross-well chaos is examined. Synchronization of the subsystems was observed for coupling strength k > 0.4. It is found that when the oscillators are operated in the regime for which two attractors coexist in phase space, basin bifurcation sequences occur leading to n + 1, n ≥ 2 basins as the coupling is varied — a signature of Wada structure and final-state sensitivity. However, in the region of complete synchronization, the basins structure is identical with that of the single oscillators and retains its essential features including fractal basin boundaries.      

强耦合混沌系统中的近似同步

以单向驱动耦合Lorenz振子一维链为研究对象，研究振子间的混沌同步行为. 数值计算结果表明，对于变量y驱动x的耦合方式，在合适的耦合强度下，会出现第一个振子和第二个振子不同步，而与次近邻非直接连接的振子(如第三个振子)近似同步. 进一步研究表明，出现这一现象的原因是在大耦合强度下，对于这种驱动方式，第一个振子和第二个振子间出现驱动单变量近似同步；虽然它们之间未出现所有变量的完全同步，但是驱动信号事实上已经传递下去了. 关键词： Lorenz振子 混沌同步 近似同步     

耦合分数阶混沌振荡器的主-从同步

近年来对分数阶系统的动力学研究得到了较为广泛的关注。本文研究了基于主-从耦合同步法的同步技术并实现了两个耦合的分数阶振荡器的混沌同步。仿真结果表明：在适当的耦合强度的调节下，该方法可实现两个耦合分数阶混沌振荡器的准确同步，且分数阶混沌振荡器的同步率明显慢于整数阶混沌振荡器的同步率；而耦合分数阶混沌振荡器在实现同步的过程中，随着阶数的提高，同步误差曲线变得平滑，这表明，系统阶数的提高改善了耦合混沌振荡器实现同步的平稳性。     

Synchronization of oscillators coupled through an environment

We study synchronization of oscillators that are indirectly coupled through their interaction with an environment. We give criteria for the stability or instability of a synchronized oscillation. Using these criteria we investigate synchronization of systems of oscillators which are weakly coupled, in the sense that the influence of the oscillators on the environment is weak. We prove that arbitrarily weak coupling will synchronize the oscillators, provided that this coupling is of the ‘right’ sign. We illustrate our general results by applications to a model of coupled GnRH neuron oscillators proposed by Khadra and Li [A. Khadra, Y.X. Li, A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons, Biophys. J. 91 (2006) 74-83.], and to indirectly weakly-coupled λ-ω oscillators.     

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic R?ssler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.     

Two scenarios of breaking chaotic phase synchronization

Two types of phase synchronization (accordingly, two scenarios of breaking phase synchronization) between coupled stochastic oscillators are shown to exist depending on the discrepancy between the control parameters of interacting oscillators, as in the case of classical synchronization of periodic oscillators. If interacting stochastic oscillators are weakly detuned, the phase coherency of the attractors persists when phase synchronization breaks. Conversely, if the control parameters differ considerably, the chaotic attractor becomes phase-incoherent under the conditions of phase synchronization break.     

Rich-club network topology to minimize synchronization cost due to phase difference among frequency-synchronized oscillators

As exemplified by power grids and large-scale brain networks, some functions of networks consisting of phase oscillators rely on not only frequency synchronization, but also phase synchronization among the oscillators. Nevertheless, even after the oscillators reach frequency-synchronized status, the phase synchronization is not always accomplished because the phase difference among the oscillators is often trapped at non-zero constant values. Such phase difference potentially results in inefficient transfer of power or information among the oscillators, and avoids proper and efficient functioning of the networks. In the present study, we newly define synchronization cost by using the phase difference among the frequency-synchronized oscillators, and investigate the optimal network structure with the minimum synchronization cost through rewiring-based optimization. By using the Kuramoto model, we demonstrate that the cost is minimized in a network with a rich-club topology, which comprises the densely-connected center nodes and low-degree peripheral nodes connecting with the center module. We also show that the network topology is characterized by its bimodal degree distribution, which is quantified by Wolfson’s polarization index.     

The effect of finite response–time in coupled dynamical systems

The paper investigates synchronization in unidirectionally coupled dynamical systems wherein the influence of drive on response is cumulative: coupling signals are integrated over a time interval τ. A major consequence of integrative coupling is that the onset of the generalized and phase synchronization occurs at higher coupling compared to the instantaneous (τ?=?0) case. The critical coupling strength at which synchronization sets in is found to increase with τ. The systems explored are the chaotic Rössler and limit cycle (the Landau–Stuart model) oscillators. For coupled Rössler oscillators the region of generalized synchrony in the phase space is intercepted by an asynchronous region which corresponds to anomalous generalized synchronization.     

Synchronization of chaotic oscillator time scales

We consider chaotic oscillator synchronization and propose a new approach for detecting the synchronized behavior of chaotic oscillators. This approach is based on analysis of different time scales in the time series generated by coupled chaotic oscillators. We show that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are particular cases of the synchronized behavior called time-scale synchronization. A quantitative measure of chaotic oscillator synchronous behavior is proposed. This approach is applied to coupled Rössler systems.     

Synchronization of an ensemble of oscillators regulated by their spatial movement

Synchronization for a collection of oscillators residing in a finite two dimensional plane is explored. The coupling between any two oscillators in this array is unidirectional, viz., master-slave configuration. Initially the oscillators are distributed randomly in space and their autonomous time-periods follow a Gaussian distribution. The duty cycles of these oscillators, which work under an on-off scenario, are normally distributed as well. It is realized that random hopping of oscillators is a necessary condition for observing global synchronization in this ensemble of oscillators. Global synchronization in the context of the present work is defined as the state in which all the oscillators are rendered identical. Furthermore, there exists an optimal amplitude of random hopping for which the attainment of this global synchronization is the fastest. The present work is deemed to be of relevance to the synchronization phenomena exhibited by pulse coupled oscillators such as a collection of fireflies.     

Synchronization of Rössler oscillators on scale-free topologies

We study the synchronization of Rössler oscillators as prototypes of chaotic systems on scale-free complex networks. As it turns out, the underlying topology crucially affects the global synchronization properties. In particular, we show that the existence of loops facilitates the synchronizability of the system, whereas Rössler oscillators do not synchronize on tree-like topologies beyond a certain size. Moreover, it is not the mere number of loops that counts for synchronization but also the type of loops. By considering Cayley trees modified by additional loops in different ways, we find out that also the distribution of shortest path lengths between two oscillators plays an important role for the global synchronization.     

Consequential noise-induced synchronization of indirectly coupled self-sustained oscillators

We consider the dynamics of identical self-sustained oscillators coupled via a common linear system (beam), which is perturbed by noise. We demonstrate that increasing the noise intensity induces complete synchronization between the oscillators and, surprisingly, their in-phase synchronization with the beam. This new phenomenon of in-phase synchronization of both the oscillators and the oscillating beam arises when the noise intensity exceeds a threshold value, and can not appear in the deterministic case where the beam stably oscillates in anti-phase with the synchronized oscillators (as it is in the case of the Huygens clocks synchronization). Similar behavior persists for slightly non-identical oscillators.     

Synchronization transition of limit-cycle system with homogeneous phase shifts

The behaviors of coupled oscillators, each of which has periodic motion with random natural frequency in the absence of coupling, are investigated when phase shifts are considered. In the system of coupled oscillators, phase shifts are the same between different oscillators. Synchronization and synchronization transition are revealed with different phase shifts. Phase shifts play an important role for this kind of system. When the phase shift α<0.5π, the synchronization state can be attained by increasing the coupling, and the system cannot reach the synchronization state while α≥q0.5π. A clear scaling between complete synchronization critical coupling strength K_pc and α-0.5π is found.     

Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect

Phase synchronization of two linearly coupled Rossler oscillators with parameter misfits is explored.It is found that depending on parameter mismatches,the synchronization of phases exhibits different manners.The synchronization regime can be divided into three regimes.For small mismatches,the amplitude-insensitive regime gives the phase-dominant synchronization; When the parameter misfit increases,the amplitudes and phases of oscillators are correlated,and the amplitudes will dominate the synchronous dynamics for very large mismatches.The lag time among phases exhibits a power law when phase synchronization is achieved.     

Synchronization in arrays of coupled self-induced friction oscillators

We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.     

Synchronization in Coupled Oscillators with Two Coexisting Attractors

Dynamics in coupled Dufling oscillators with two coexisting symmetrical attractors is investigated. For a pair of Dufl~ng oscillators coupled linearly, the transition to the synchronization generally consists of two steps： Firstly, the two oscillators have to jump onto a same attractor, then they reach synchronization similarly to coupled monostable oscillators. The transition scenarios to the synchronization observed are strongly dependent on initial conditions.     

耦合振子系统的多稳态同步分析

本文讨论了一维闭合环上Kuramoto相振子在非对称耦合作用下同步区域出现的多定态现象. 研究发现在振子数N≤3情形下系统不会出现多态现象, 而N≥4多振子系统则呈现规律的多同步定态. 我们进一步对耦合振子系统中出现的多定态规律及定态稳定性进行了理论分析, 得到了定态渐近稳定解. 数值模拟多体系统发现同步区特征和理论描述相一致. 研究结果显示在绝热条件下随着耦合强度的减小, 系统从不同分支的同步态出发最终会回到同一非同步态. 这说明, 耦合振子系统在非同步区由于运动的遍历性而只具有单一的非同步态, 在发生同步时由于遍历性破缺会产生多个同步定态的共存现象.