title: Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis.
abstract: Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find all of the equilibria for various choices of coupling constants K, natural frequencies, and on different graphs. We note that for even modest sizes (N ~ 10-20), the number of equilibria is already more than 100 000. We analyze the stability of each computed equilibrium as well as the configuration of angles. Our exploration of the equilibrium landscape leads to unexpected and possibly surprising results including non-monotonicity in the number of equilibria, a predictable pattern in the indices of equilibria, counter-examples to conjectures, multi-stable equilibrium landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equilibrium distribution as a function of the number of cycles in the graph.
authors.affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556, USA.
authors.forename: Dhagash
authors.initials: D
authors.lastname: Mehta
authors.affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695, USA.
authors.forename: Noah S
authors.initials: NS
authors.lastname: Daleo
authors.affiliation: Automatic Control Laboratory, Swiss Federal Institute of Technology (ETH) Zürich, 8092 Zürich, Switzerland.
authors.forename: Florian
authors.initials: F
authors.lastname: Dörfler
authors.affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556, USA.
authors.forename: Jonathan D
authors.initials: JD
authors.lastname: Hauenstein