Abstract

When does a system of coupled oscillators synchronize? This central question in dynamical systems arises in applications ranging from power grids to neuroscience to biology: why do fireflies sometimes begin flashing in harmony? Perhaps the most studied model is due to Kuramoto (1975); we  analyze the Kuramoto model from the perspectives of algebra and topology. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks); our work also tackles more general situations. We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. The talk will include a surprising (at least to us!) connection to Segre varieties, and close with examples of computations using the Macaulay2 software package “Oscillator” Joint work with Heather Harrington (Oxford/Dresden) and Mike Stillman (Cornell).