Parametric Resonance in Time-Dependent Networks of Hopf Oscillators

We study a collection of limit cycle oscillators with different frequencies interacting via networks. With static networks, our specific class of models produces a frequency learning mechanism leading the collection of oscillators to ultimately adopt a single consensual frequency which, once reached, subsists even if interactions are switched off (i.e. interactions produce a plastic deformation of the dynamics). The consensual state is asymptotically reached independent of the networks’ topologies. However, the topologies affect the convergence rate. For time-dependent networks, new dynamical patterns may emerge. These produce a time-dependent spectrum of the Laplacian matrices describing the algebraic connectivities. For time-dependent circulant networks, we are able to show that linear stability analysis in the vicinity of the stationary consensual state produces Hill type equations (i.e a second-oder linear ODEs with oscillating parameters), thus offering the possibility of destabilization via parametric resonance phenomena.