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arXiv · 2312.16069

Algebraic aspects of homogeneous Kuramoto oscillators

Heather Harrington·Hal Schenck·Mike Stillman
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Abstract

We investigate algebraic and topological signatures of networks of coupled oscillators. 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). 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. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states.

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Heather Harrington, Hal Schenck, Mike Stillman. 2025-01-04. Algebraic aspects of homogeneous Kuramoto oscillators. https://arxiv.org/abs/2312.16069

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