Graph partitions and cluster synchronization in networks of oscillators

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators

Complex dynamics in adaptive phase oscillator networks

Networks of coupled dynamical units give rise to collective dynamics such as the synchronization of oscillators or neurons in the brain. The ability of the network to adjust coupling strengths between units in accordance with their activity arises naturally in a variety of contexts, including neural plasticity in the brain, and adds an additional layer of complexity: the dynamics on the nodes influence the dynamics of the network and vice versa. We study a model of Kuramoto phase oscillators with a general adaptive learning rule with three parameters (strength of adaptivity, adaptivity offset, adaptivity shift). This rule includes as special cases learning paradigms such as (anti-)Hebbian learning and spike time dependent plasticity (STDP). Importantly, the adaptivity parameter allows to study the impact of adaptation on the collective dynamics as we move away from the non-adaptive case given by stationary coupling. First, we carry out a detailed bifurcation analysis for N = 2 oscillators with (un-)directed coupling strengths. Adaptation dynamics in terms of nontrivial bifurcations arises only when the strength of adaptation exceeds a critical threshold. Whereas the paradigms of (anti-)Hebbian learning and STDP result in non-trivial multi-stability and bifurcation scenarios, mixed-type learning rules exhibit even more complicated and rich dynamics including a period doubling cascade to chaotic dynamics as well as oscillations displaying features of both librational and rotational character. Second, we numerically investigate a larger system with N = 50 oscillators and explore dynamic similarities with the case of N = 2 oscillators

Partial Phase Cohesiveness in Networks of Communitinized Kuramoto Oscillators

Partial synchronization of neuronal ensembles are often observed in the human brain, which is believed to facilitate communication among anatomical regions demanded by cognitive tasks. Since such neurons are commonly modeled by oscillators, to better understand their partial synchronization behavior, in this paper we study community-driven partial phase cohesiveness in networks of communitinized Kuramoto oscillators, where each community itself consists of a population of all-to-all coupled oscillators. Sufficient conditions on the algebraic connectivity of the selected communities are obtained to guarantee the appearance of their phase cohesiveness, while leaving the remaining communities incoherent. These conditions are further reduced to the form of the lower bounds on the coupling strengths for the connections linking the selected communities. We also show that the ultimate level of the phase cohesiveness that the oscillators asymptotically converge to is predictable. Finally, numeral studies are performed to validate the obtained results

Synchronisation phenomena with time delay

I study a simple model of synchronisation proposed by Jensen (2008). The relevant degrees of freedom are expected to be strictly increasing functions of time, such as the total angle swept out by an oscillator. The model is rooted in Winfree’s mean-field model for spontaneous synchronisation; some of Winfree’s basic assumptions, such as identical or nearly identical dynamics and identical couplings, are therefore retained. I investigated the behaviour of the present model with respect to synchronisation without and in the presence of time delay. The mathematical treatment focuses on characterising the synchronised state as either at- tractive or repulsive, producing a theory (which ultimately leads to a phase diagram) that compares well with numerics. I employed a perturbative approach, linearising in small time delays and small phase differences. The interaction between individual oscillators is captured by an interaction matrix, which does not require further approximation, i.e. lattice structure enters exactly. To link with established results in the literature, a mean field theory, however, is also studied. The main result is that these typically systems synchronise due to a time delay.Open Acces

The Kuramoto model in complex networks

181 pages, 48 figures. In Press, Accepted Manuscript, Physics Reports 2015 Acknowledgments We are indebted with B. Sonnenschein, E. R. dos Santos, P. Schultz, C. Grabow, M. Ha and C. Choi for insightful and helpful discussions. T.P. acknowledges FAPESP (No. 2012/22160-7 and No. 2015/02486-3) and IRTG 1740. P.J. thanks founding from the China Scholarship Council (CSC). F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grants No. 2011/50761-2 and No. 2013/26416-9) for financial support. J.K. would like to acknowledge IRTG 1740 (DFG and FAPESP).Peer reviewedPreprin

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com