A new approach to bifurcations in the Kuramoto model
The author considers the Kuramoto model [\textit{Y. Kuramoto}, Chemical oscillations, waves, and turbulence. Berlin etc.: Springer-Verlag (1984; Zbl 0558.76051)] for the dynamics of networks of coupled oscillators. This model has found widespread application in the study of synchronisation, as reported in [\textit{S. H. Strogatz}, Physica D 143, No. 1--4, 1--20 (2000; Zbl 0983.34022)]. Here, bifurcations involving phase-locked solutions -- those for which all oscillators rotate at a common frequency with fixed angular differences -- on a ring network are investigated. The author derives criteria for a branch of stable phase-locked solutions to collide with a branch of \(1\)-saddles as the coupling parameter is varied, and illustrates those in specific examples. Finally, he studies the bistability phenomenon, i.e., the existence of two distinct stable phase-locked solutions, as well as higher-order bifurcations. His analysis is largely based on a novel vector flow which yields a local parametrisation of the governing equations close to the bifurcation regime.
scientific article; zbMATH DE number 7358698
scientific article