Topological phase transition in the periodically forced Kuramoto model

• Published in: Chaos, solitons and fractals solitons and fractals, 2021-04, Vol.145, p.110816, Article 110816
• Main Authors: Wright, E.A.P., Yoon, S., Mendes, J.F.F., Goltsev, A.V.
• Format: Article
• Language: English
• Subjects: Bifurcation analysis; Entrainment; Singular point; Synchronization; Topological transition; Winding number
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2021.110816

•We reveal a topological phase transition in the periodically forced Kuramoto model.•The transition separates states with oscillating and rotating collective dynamics.•The critical point is accompanied by an abrupt drop in the average cycling frequency.•Oscillating and rotating states have distinct winding numbers.•Winding numbers are defined relative to a singular point in the order-parameter space. A complete bifurcation analysis of explicit dynamical equations for the periodically forced Kuramoto model was performed in [L. M. Childs and S. H. Strogatz. Chaos 18, 043128 (2008)], identifying all bifurcations within the model. We show that the phase diagram predicted by this analysis is incomplete. Our numerical analysis of the equations reveals that the model can also undergo an abrupt phase transition from oscillations to wobbly rotations of the order parameter under increasing field frequency or decreasing field strength. This transition was not revealed by bifurcation analysis because it is not caused by a bifurcation, and can neither be classified as first nor second order since it does not display critical phenomena characteristic of either transition. We discuss the topological origin of this transition and show that it is determined by a singular point in the order-parameter space.