Networks of Limit Cycle Oscillators with Parametric Learning Capability

Abstract

The fundamental idea that synchronized patterns emerge in networks of interacting oscillators is revisited by allowing a parametric learning mechanism to operate on the local dynamics. The local dynamics consist of stable limit cycle oscillators which, due to mutual interactions, are allowed, via an adaptive process, to permanently modify their frequencies. Adaptivity is made possible by conferring to each oscillator’s frequency the status of an additional degree of freedom. The network of individual oscillators is ultimately driven to a stable synchronized oscillating state which, once reached, survive even if mutual interactions are removed. Such a permanent, plastic type deformation of an initial to a final consensual state is realized by a dissipative mechanism which vanishes once a consensus is established. By considering diffusive couplings between position- and velocity-dependent state variables, we are able to analytically explore the resulting dynamics and in particular to calculate the resulting consensual state. The ultimate consensual state is topology network-independent. However, the interplay between the graph connectivity and the local dynamics does strongly influence the learning rate.