Dynamics on networks of cluster states for globally coupled phase oscillators

Copyright © by Society for Industrial and Applied Mathematics. Unauthorized reproduction of this article is prohibited. Its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. We investigate such a heteroclinic network between partially synchronized states where the phases cluster into three groups. For the coupling considered there exist 30 different three-cluster states in the case of five oscillators. We study the structure of the heteroclinic network and demonstrate that it is possible to navigate around the network by applying small impulsive inputs to the oscillator phases. This paper shows that such navigation may be done reliably even in the presence of noise and frequency detuning, as long as the input amplitude dominates the noise strength and the detuning magnitude, and the time between the applied pulses is in a suitable range. Furthermore, we show that, by exploiting the heteroclinic dynamics, frequency detuning can be encoded as a spatiotemporal code. By changing a coupling parameter we can stabilize the three-cluster states and replace the heteroclinic network by a network of excitable three-cluster states. The resulting “excitable network” has the same structure as the heteroclinic network and navigation around the excitable network is also possible by applying large impulsive inputs. We also discuss features that have implications for related models of neural activity

Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators

This is a preprint of an article whose final and definitive form has been published in DYNAMICAL SYSTEMS © 2005 copyright Taylor & Francis; DYNAMICAL SYSTEMS is available online at: http://www.informaworld.com/openurl?genre=article&issn=1468-9367&volume=20&issue=3&spage=333Systems of coupled oscillators may exhibit spontaneous dynamical formation of attracting synchronized clusters with broken symmetry; this can be helpful in modelling various physical processes. Analytical computation of the stability of synchronized cluster states is usually impossible for arbitrary nonlinear oscillators. In this paper we examine a particular class of strongly nonlinear oscillators that are analytically tractable. We examine the effect of isochronicity (a turning point in the dependence of period on energy) of periodic oscillators on clustered states of globally coupled oscillator networks. We extend previous work on networks of weakly dissipative globally coupled nonlinear Hamiltonian oscillators to give conditions for the existence and stability of certain clustered periodic states under the assumption that dissipation and coupling are small and of similar order. This is verified by numerical simulations on an example system of oscillators that are weakly dissipative perturbations of a planar Hamiltonian oscillator with a quartic potential. Finally we use the reduced phase-energy model derived from the weakly dissipative case to motivate a new class of phase-energy models that can be usefully employed for understanding effects such as clustering and torus breakup in more general coupled oscillator systems. We see that the property of isochronicity usefully generalizes to such systems, and we examine some examples of their attracting dynamics

Multi-cluster dynamics in coupled phase oscillator networks

In this paper we examine robust clustering behaviour with multiple nontrivial clusters for identically and globally coupled phase oscillators. These systems are such that the dynamics is completely determined by the number of oscillators N and a single scalar function $g(\varphi)$ (the coupling function). Previous work has shown that (a) any clustering can stably appear via choice of a suitable coupling function and (b) open sets of coupling functions can generate heteroclinic network attractors between cluster states of saddle type, though there seem to be no examples where saddles with more than two nontrivial clusters are involved. In this work we clarify the relationship between the coupling function and the dynamics. We focus on cases where the clusters are inequivalent in the sense of not being related by a temporal symmetry, and demonstrate that there are coupling functions that give robust heteroclinic networks between periodic states involving three or more nontrivial clusters. We consider an example for N=6 oscillators where the clustering is into three inequivalent clusters. We also discuss some aspects of the bifurcation structure for periodic multi-cluster states and show that the transverse stability of inequivalent clusters can, to a large extent, be varied independently of the tangential stability

Winnerless Competition in Neural Dynamics; Cluster Synchronisation of Coupled Oscillators

Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. Such a heteroclinic network is generated, where the phases cluster into three groups, within a specific regime of parameters when the phase oscillators are globally coupled using the function $g(\varphi) = -\sin(\varphi + \alpha) + r \sin(2\varphi + \beta)$. The resulting network switches between 30 partially synchronised states for a system of $N=5$ oscillators. Considering the states that are visited and the time spent at those states a spatio-temporal code can be generated for a given navigation around the network. We explore this phenomenon further by investigating the effect that noise has on the system, how this system can be used to generate a spatio-temporal code derived from specific inputs and how observation of a spatio-temporal code can be used to determine the inputs that were presented to the system to generate a given coding. We show that it is possible to find chaotic attractors for certain parameters and that it is possible to detail a genetic algorithm that can find the parameters required to generate a specific spatio-temporal code, even in the presence of noise. In closing we briefly explore the dynamics where $N>5$ and discuss this work in relation to winnerless competition.EPSR

Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators

A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.Comment: 26 pages, 3 figure

Experimental observation of chimera and cluster states in a minimal globally coupled network

A "chimera state" is a dynamical pattern that occurs in a network of coupled identical oscillators when the symmetry of the oscillator population is broken into synchronous and asynchronous parts. We report the experimental observation of chimera and cluster states in a network of four globally coupled chaotic opto-electronic oscillators. This is the minimal network that can support chimera states, and our study provides new insight into the fundamental mechanisms underlying their formation. We use a unified approach to determine the stability of all the observed partially synchronous patterns, highlighting the close relationship between chimera and cluster states as belonging to the broader phenomenon of partial synchronization. Our approach is general in terms of network size and connectivity. We also find that chimera states often appear in regions of multistability between global, cluster, and desynchronized states

Emergence of multicluster chimera states

We thank Prof. L. Huang for helpful discussions. This work was partially supported by ARO under Grant No. W911NF-14-1-0504 and by NSF of China under Grant No. 11275003. The visit of NY to Arizona State University was partially sponsored by Prof. Z. Zheng and the State Scholarship Fund of China.Peer reviewedPublisher PD