Finite-size-induced transition in ensemble of globally coupled oscillators

The collective behavior of overdamped nonlinear noise-driven oscillators coupled via mean field is investigated numerically. When a coupling constant is increased, a transition in the dynamics of the mean field is observed. This transition scales with the number of oscillators and disappears when this number tends to infinity. Analytical arguments explaining the observed scaling are presented.     

Controlling synchronization in an ensemble of globally coupled oscillators

We propose a technique to control coherent collective oscillations in ensembles of globally coupled units (self-sustained oscillators or maps). We demonstrate numerically and theoretically that a time delayed feedback in the mean field can, depending on the parameters, enhance or suppress the self-synchronization in the population. We discuss possible applications of the technique.     

Long-range correlations and ensemble inequivalence in a generalized ABC model

A generalization of the ABC model, a one-dimensional model of a driven system of three particle species with local dynamics, is introduced, in which the model evolves under either (i) density-conserving or (ii) nonconserving dynamics. For equal average densities of the three species, both dynamical models are demonstrated to exhibit detailed balance with respect to a Hamiltonian with long-range interactions. The model is found to exhibit two distinct phase diagrams, corresponding to the canonical (density-conserving) and grand canonical (density nonconserving) ensembles, as expected in long-range interacting systems. The implications of this result to nonequilibrium steady states, such as those of the ABC model with unequal average densities, are briefly discussed.     

一个全局耦合不连续映像格子中的冻结化随机图案模式

本文研究了一类既不连续又不可逆分段线性映像构成的全局耦合映像格子系统中的一类典型集体动力学行为, 即冻结化随机图案模式. 计算了平均同步序参量和最大李雅普诺夫指数随耦合强度的变化. 结果显示, 当耦合强度超过某个阈值后, 在给定动力学变量的初始下, 系统几乎都能达到完全或部分同步状态, 出现冻结化随机图案. 这些现象表明, 耦合映像格子系统中存在着多个共存的吸引子. 因此, 其冻结化图案的结构和分布敏感地依赖于格点动力学变量初始值的选取. 感兴趣地是, 即使当单映像处于混沌状态时, 格点间的耦合仍能将系统调制到规则的运动状态, 这种特征对于混沌控制具有重要的利用价值. 上述丰富动力学行为的出现是由于单映像中不连续性和不可逆性相互作用的结果.     

Synchronization in a system of globally coupled oscillators with time delay

We study the synchronization phenomena in a system of globally coupled oscillators with time delay in the coupling. The self-consistency equations for the order parameter are derived, which depend explicitly on the amount of delay. Analysis of these equations reveals that the system in general exhibits discontinuous transitions in addition to the usual continuous transition, between the incoherent state and a multitude of coherent states with different synchronization frequencies. In particular, the phase diagram is obtained on the plane of the coupling strength and the delay time, and ubiquity of multistability as well as suppression of the synchronization frequency is manifested. Numerical simulations are also performed to give consistent results.     

Resonance tongues in a system of globally coupled FitzHugh-Nagumo oscillators with time-periodic coupling strength

We investigate the dynamics of a population of globally coupled FitzHugh-Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.     

Clustering zones in the turbulent phase of a system of globally coupled chaotic maps

The paper develops an approach to investigate the clustering phenomenon in the system of globally coupled chaotic maps first introduced by Kaneko in 1989. We obtain a relation between the transverse and longitudinal multipliers of the periodic clusters and prove the stability of these clusters for the case of symmetric, equally populated distributions between subclusters. Stable clusters emanate from the periodic windows of the logistic map and extend far into the turbulent phase. By numerical simulations we estimate a total basin volume of low-periodic clusters issued from the period-3 window and analyze the basin structure. The complement to the basin volume is ascribed to chaotic, very asymmetric high-dimensional clusters that are characterized by the presence of one or more leading clusters, accumulating about half of the oscillators while all the remaining oscillators do not cluster at all.     

Bifurcations in globally coupled map lattices

The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius-Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The bifurcation behavior of coupled tent maps near the chaotic band merging point is presented. Furthermore, the time-independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations     

一个全局耦合不连续映像格子中的奇异态

文章研究了一类由既不可逆又不连续映像构成的全局耦合映像格子系统中的奇异态行为,计算了系统的同步序参量和空间振幅变化图.结果表明,在某些特定的参数区间内,耦合映像格子系统会出现奇异态或团簇态,并且敏感地依赖于耦合强度的选择.上述丰富的动力学现象是由于单映像中不连续、不可逆性以及空间耦合相互作用的结果.通过数值模拟找到了奇异态或团簇态出现的特定参数区域.     

Differentiation in a globally coupled circle map with growth and death

A key characteristic of biological systems is the continuous life cycle where cells are born, grow and die. From a dynamical point of view the events of cell division and cell death are of paramount importance and constitute a radical departure from systems with a fixed size. In this paper, a globally coupled circle map where elements can dynamically be added and removed is investigated for the conditions under which differentiation of roles can occur. In the presence of an external source, it is found that populations of very long-living cells are sustained by short-living cells. In the case without an external source, it is found that at higher nonlinearities of the local map, large populations cannot be sustained with a previously employed division strategy but that a different and conceptually equally natural division strategy allows for differentiation of roles.     

Transitivity and blowout bifurcations in a class of globally coupled maps

A class of globally coupled one dimensional maps is studied. For the uncoupled one dimensional map it is possible to compute the spectrum of Liapunov exponents exactly, and there is a natural equilibrium measure (Sinai–Ruelle–Bowen measure), so the corresponding `typical' Liapunov exponent may also be computed. The globally coupled systems thus provide examples of blowout bifurcations in arbitrary dimension. In the two dimensional case these maps have parameter values at which there is a transitive (topological) attractor which is a filled-in quadrilateral and, simultaneously, the synchronized state is a Milnor attractor.     

Clusters and switchers in globally coupled photochemical oscillators

We experimentally investigate the transition to synchronization in a population of photochemical oscillators with weak global coupling. Above a critical coupling strength the oscillators join a one-phase group or two-phase clusters. The number of oscillators in each cluster depends on the initial phase distribution, and irregular switching of oscillators between clusters is observed. The fully synchronized state emerges above a second critical coupling strength. In agreement with earlier theory, the experiments demonstrate the importance of population heterogeneity in cluster multistability.     

Synchronized family dynamics in globally coupled maps

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many stable solutions, except for sufficiently large coupling strengths, where the lattice elements always synchronize, and for small map parameter, where only simple fixed points are found. For smaller varepsilon and larger a, there is a portion of the parameter plane in which chaotic, asynchronous lattices are found. Over much of the parameter plane, lattices converge to states in which the maps are partitioned into a number of synchronized families. The dynamics and stability of two-family states (solutions partitioned into two families) are explored in detail. (c) 1999 American Institute of Physics.     

Phase clustering in globally coupled photochemical oscillators

We experimentally investigate the formation of clusters in a population of globally coupled photochemical oscillators. The system consists of catalytic micro-particles in Belousov-Zhabotinsky solution and the coupling exploits the excitatory properties of light; an increase in the light intensity leads to excitation (“firing") of an oscillator. As the coupling strength is increased, a transition occurs from incoherence to clustering, whereby the oscillators split into synchronised groups, to complete synchronisation. Multistability is observed between a one-phase cluster (fully synchronised group) and two-phase clusters (two groups with the same frequency but different phases). The results are reproduced in simulations and we demonstrate that the heterogeneity of the population as well as the relaxational nature of the oscillators is important in the observation of clusters. We also examine the exploitation of the phase model for the prediction of clusters in experiments.     

Extensive and subextensive chaos in globally coupled dynamical systems

Using a combination of analytical and numerical techniques, we show that chaos in globally coupled identical dynamical systems, whether dissipative or Hamiltonian, is both extensive and subextensive: their spectrum of Lyapunov exponents is asymptotically flat (thus extensive) at the value λ(0) given by a single unit forced by the mean field, but sandwiched between subextensive bands containing typically O(logN) exponents whose values vary as λ?λ(∞)+c/logN with λ(∞)≠λ(0).     

Generating macroscopic chaos in a network of globally coupled phase oscillators

We consider an infinite network of globally coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.     

Towards a quasi-periodic mean field theory for globally coupled oscillators

We show how a quasi-periodic mean field theory may be used to understand the chaotic dynamics and geometry of globally coupled complex Ginzburg-Landau equations. The Poincaré map of the mean field equations appears to have saddlenode-homoclinic bifurcations leading to chaotic motion, and the attractor has the characteristic ρ shape identified by numerical experiments on the full equations.