Synchronization properties of network motifs: Influence of coupling delay and symmetry

Abstract

We investigate the effect of coupling delays on the synchronization properties of several network motifs. In particular, we analyze the synchronization patterns of unidirectionally coupled rings, bidirectionally coupled rings, and open chains of Kuramoto oscillators. Our approach includes an analytical and semianalytical study of the existence and stability of different in-phase and out-of-phase periodic solutions, complemented by numerical simulations. The delay is found to act differently on networks possessing different symmetries. While for the unidirectionally coupled ring the coupling delay is mainly observed to induce multistability, its effect on bidirectionally coupled rings is to enhance the most symmetric solution. We also study the influence of feedback and conclude that it also promotes the in-phase solution of the coupled oscillators. We finally discuss the relation between our theoretical results on delay-coupled Kuramoto oscillators and the synchronization properties of networks consisting of real-world delay-coupled oscillators, such as semiconductor laser arrays and neuronal circuits.

Delay stabilization of periodic orbits in coupled oscillator systems

“…From a mathematical point of view, generalizations to n oscillators and thus n-fold symmetries may be an interesting direction to pursue (D'Huys et al 2008;Choe et al submitted). For the Pyragas control scheme of planar in-phase circular orbits, it has been shown (Fiedler 2008) that only orbits whose real Floquet multipliers μ obey μ < exp(9) can be stabilized.…”

Amplitude and phase dynamics in oscillators with distributed-delay coupling

“…A significant amount of research has been carried out over the years on the analysis of phase oscillators with different types of time-delayed coupling [2,[24][25][26][27][28][29][30][31][32]. Such systems of phase oscillators demonstrate a rich variety of dynamical regimes, including chaos, synchronization, splay and chimera states, characterized by the coexistence of coherent and incoherent states.…”

Cluster synchronization and isolated desynchronization in complex networks with symmetries

“…These anecdotal studies illustrate the interesting types of CS that can occur, and suggest a broader relationship between the network structure and synchronization patterns. Recent studies have begun to draw a connection between network symmetry and CS, although all have considered simple networks where the symmetries are apparent by inspection [12][13][14] . More in-depth studies have been done involving bifurcation phenomena and synchronization in ring and point symmetry networks 15,16 .…”