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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- May 01, 2015
- Source ID
- 10.1063/1.4919696
Entities
People
- Dhagash Mehta
- Florian Dorfler
- Jonathan D Hauenstein
- Noah S. Daleo
Organizations
- Defense Advanced Research Projects Agency
- ETH Zurich
- National Science Foundation
- North Carolina State University
- University of Notre Dame