Dynamics of Minimal Networks *

Dynamics of a network of oscillators has received immense attention due to its potential application in various fields such as power grids, neuronal networks, seizure dynamics, and thermoacoustic interaction in can or can-annular type gas turbines. The stability and dynamical behavior of such networks depend on various factors including the number of oscillators, coupling parameters, and topology. These factors play a vital role in networks with low number of oscillators. Here, we study the susceptibility of minimal networks of candle-flame oscillators to the variation of the aforementioned factors. We report the presence of various dynamical states such as chimera, weak chimera, clustering, amplitude death, and partial amplitude death in both open-loop and closed-loop networks of oscillators. At short separations, the oscillators exhibit in-phase synchronization or amplitude death irrespective of their topology, whereas, at larger separations, they exhibit various states with varying probabilities of occurrence. Synchronization of oscillators is found to be easier in closed-loop compared to open-loop topologies. Further, the interaction of odd number of oscillators in a ring network prefers the occurrence of symmetry-breaking states, while networks with even number of oscillators prefer synchronization states.