Completely Degenerate Equilibria of the Kuramoto Model on Networks

TL;DR

This paper investigates completely degenerate equilibria in Kuramoto networks, proving their existence conditions based on graph structure and intrinsic frequencies, and demonstrating their instability regardless of parameters.

Contribution

It characterizes the existence and stability of completely degenerate equilibria in Kuramoto models, linking them to graph bipartiteness and Euler circuits, which is a novel insight.

Findings

Completely degenerate equilibria exist only for bipartite graphs with certain frequency conditions.

Such equilibria are always unstable regardless of parameters.

Existence of these equilibria is rare and depends on specific graph properties.

Abstract

Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this case linearizing the system at the equilibrium leads to a Jacobian matrix which is zero in every entry. We call these equilibria completely degenerate. We prove that they exist for certain intrinsic frequencies if and only if the underlying graph is bipartite, and that they do not exist for generic intrinsic frequencies. In the case of zero intrinsic frequencies, we prove that they exist if and only if the graph has an Euler circuit such that the number of steps between any two visits at the same vertex is a multiple of 4. The simplest example is the cycle graph with 4 vertices. We prove that graphs with this property exist for every number of vertices N larger than 5 and that they become asymptotically rare for N large. Regarding stability, we prove that for any choice of intrinsic frequencies, any coupling strength and any graph with at least one edge, completely degenerate equilibria are not Lyapunov stable. As a corollary, we obtain that stable equilibria in Kuramoto Networks must have at least one strictly negative eigenvalue.