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

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