Time Delayed Interactions in Networks of Self-Adapting Hopf Oscillators

A network of coupled limit cycle oscillators with delayed interactions is considered. The parameters characterizing the oscillator's frequency and limit cycle are allowed to self-adapt. Adaptation is due to time-delayed state variables that mutually interact via a network. The self-adaptive mechanisms ultimately drive all coupled oscillators to a consensual cyclo-stationary state where the values of the parameters are identical for all local systems. They are analytically expressible. The interplay between the spectral properties of the coupling matrix and the time-delays determine the conditions for which convergence towards a consensual state takes place. Once reached, this consensual state subsists even if interactions are removed. In our class of models, the consensual values of the parameters neither depend on the delays nor on the network's topology.