Delay-coupled phase oscillators on a star network: the effect of degree inhomogeneity
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Delay-coupled phase oscillators on a star network: the effect of degree inhomogeneity Umeshkanta Singh Thounaojam a Department of Physics, Shivaji College, University of Delhi, New Delhi 110067, India
Received 25 October 2019 / Received in final form 9 April 2020 Published online 13 July 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. We study the effect of nonuniform delay coupling due to degree inhomogeneity in a system of identical Kuramoto oscillators on a bipartite network with the star topology. The hub and peripheral oscillators have synchronized solutions with phase shift between them. Transcendental equations are obtained for the collective frequencies and phase shifts, and linear stability conditions are derived for the solutions. Numerical simulations are in good agreement with these results. Systems in which collective frequencies depend on both delays and phase shifts are important in the context of optimizing synchronization rates.
1 Introduction Efforts to understand how synchrony arises through detailed studies of interacting oscillators is an active area of research [1], given the ubiquity of synchronization in a wide variety of coupled natural systems spanning the physical, biological, chemical, and social sciences [2–5]. One of the most widely studied such models has been due to Kuramoto [6], who considered globally coupled phase oscillators having different intrinsic frequencies. This model offers a bridge between dynamics and statistical mechanics, since there is essentially a second-order phase transition to global synchrony in the system as a function of the coupling strength [7]. In this paper we consider phase oscillators coupled on a network with the star topology, as shown in Figure 1. The basic dynamics is taken to be that in the Kuramoto model, but with two additional effects in the coupling: time-delay and degree inhomogeneity. The incorporation of time-delay in the coupling is motivated by the fact that in natural systems, signals are transmitted with a finite velocity and thus there is a natural time-delay in the processing of information between the constituent units. Delays in signal transmission also have an impact on selforganization of coupled oscillators [8–13]. With time-delay there can be multistabiltity, and as shown by Schuster and Wagner [8] who investigated the effect of time-delay on the mutual entrainment of two coupled phase oscillators, multiple synchronized solutions coexist. The stability criterion for synchronized solutions when there was time-delay coupling was subsequently derived by Earl and Strogatz [9]. Inhomogeneous coupling is likely to be a feature that a
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should be considered in realistic models. Particularly in the case of scale-free networks [14] where the degree distribution has a large variance, all the oscillators may not be coupled uniformly. Hubs in the network receive signals from a large numb
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