A global bifurcation organizing rhythmic activity in a coupled network

We study a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand.

Automated summary

In this study, we investigate a system of coupled phase oscillators driven by random intrinsic frequencies near a saddle-node on invariant circle bifurcation. The system undergoes a phase transition and changes its qualitative properties of collective dynamics under the variation of control parameters. By using Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we demonstrate that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation and noncontractible ones after bifurcation. Both families are stable in the model at hand.