Designing the Dynamics of Coupled Oscillators

Abstract

By designing coupling to control populations of oscillators, we can control their synchonisation behaviour. Oscillators (e.g. neurons) can be coupled on different levels. The most basic level is through links between pairs of oscillators. However, using graphs with only pairwise links is not necessarily a satisfactory approximation of reality as nonpairwise interactions can be found in many dynamical systems including social networks and the human brain. Even though the effects of these nonpairwise interactions have been observed, described and modeled in a wide range of oscillatory systems, controlling nonpairwise interactions in arbitrary populations of oscillators has remained a relatively unexplored area. In this thesis we generalise synchronisation engineering to control nonpairwise interactions in arbitrary systems. We designed a nonlinear time-delayed coupling that can be used to match the phase reduction of a system of oscillators to a target phase model. The contribution of this thesis is allowing for nonpairwise interactions in the target phase model. We used an optimisation proceidure to find coupling parameters to match a nonpairwise target phase model that has the collective behaviour we aim to introduce to the system We found that we need one additional filter to find the parameter sets that match the bifurcation of both in-phase and splay configuration in to the nonpairwise target phase model.