Dynamics of Reaction-Diffusion Oscillators in Star Networks

We experimentally and theoretically investigate the dynamics of inhibitory coupled self-driven oscillators on a star network in which a single central hub node is connected to $k$ peripheral arm nodes. The system consists of water-in-oil Belousov-Zhabotinsky $\sim$100$\mu$m emulsion drops contained in storage wells etched in silicon wafers. We observed three dynamical attractors by varying the number of arms in the star graph and the coupling strength; ($i$) unlocked; uncorrelated phase shifts between all oscillators, ($ii$) locked; arm-hubs synchronized in-phase with a $k$-dependent phase shift between the arm and central hub, and ($iii$) center silent; central hub stopped oscillating and the arm-hubs oscillated without synchrony. We compare experiment to theory. For case ($ii$), we identified a logarithmic dependence of the phase shift on star degree, and were able to discriminate between contributions to the phase shift arising from star topology and oscillator chemistry.