Synchronization in phase oscillator networks with 'ring' and 'small world' link topologies and different dependences of the oscillator frequency on its network location

Background and Objectives: In this paper, we consider the general case of the establishment/destruction of a completely synchronous state of phase oscillator networks with topologies of links such as “ring” and “small world”. The natural frequencies of the node oscillators are supposed to be distributed along the network according to an arbitrary law. Materials and Methods: The network of Kuramoto oscillators, consisting of 1000 node elements, was considered as the system under study theoretically and numerically. Results: The influence of quantitative characteristics of the dependence of natural frequencies on the number (spatial coordinate) of the oscillator on the onset of a completely synchronous state of the network of phase oscillators as well as the mechanism of the transition to a completely synchronous regime has been studied. An analytical expression forthe critical value ofthe coupling parameter correspondingtothe establishment of a fully synchronous regime within the network under consideration has been deduced. The theoretical results obtained have been compared with the results of the direct numerical simulation of the oscillator network behavior, with the excellent agreement being observed. Conclusion: It has been found that the dependence of the natural frequencies of oscillators on the spatial coordinate (or, on the number of the oscillator in the network) in the case of networks with the topology of links such as “ring” and “small world” determines completely the properties of such networks from the point of view of establishing the phase synchronization. Having based on the “ring” and “small world” network properties, it is possible to solve not only the problem of finding the critical value of the coupling parameter for the known frequency dependences on the coordinate, but also the problem of synthesizing such networks with the predetermined properties.