Coarse graining the dynamics of delayed phase oscillators on Cayley trees by star networks

Abstract

It is of interest in many natural and artificial networks to find simpler systems to understand spatio-temporal dynamics. For this, we consider an extended fragmentary networks by considering a system of phase oscillators with time delay and degree heterogeneity on Cayley trees to address it. The coupled oscillators on Cayley trees exhibit various dynamical regimes such as clusters synchronization, desynchronization, and phase-flip between synchronized clusters. The dynamical regimes and transitions of oscillators on the Cayley tree are found to be similar to the star network’s oscillator dynamics. We confirm the Cayley tree and star network’s functional equivalence by finding the collective frequencies of phase oscillators in both systems. Analytical estimates of linearly stable common frequencies of oscillators on star networks agree with oscillator’s numerical solutions on Cayley trees. Our results hold for identical and non-identical oscillators, and they provide a theoretical framework to understand the collective dynamics of delayed phase oscillators in extended systems.