Codimension-two bifurcations induce hysteresis behavior and multistabilities in delay-coupled Kuramoto oscillators

Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott–Antonsen’s manifold, two-parameter bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values; thus, there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the MATLAB package DDE-BIFTOOL, the location of Bautin and double Hopf points and detailed dynamics are theoretically determined. We find that, near these critical points, four coherent states (two of which are stable) and a stable incoherent state may coexist and that the system undergoes Neimark–Sacker bifurcation of periodic solutions. Finally, the clear scenarios about the synchronous transition in delayed Kuramoto model are depicted.