A global bifurcation organizing rhythmic activity in a coupled network

Abstract

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.

Social and nonlinear dynamics unite: Musical group synchrony

“…2) Hierarchical models that capture relationships among subgroups should be developed for larger musical groups such as bands and orchestras. Recent extensions of hierarchical autoregressive models [31] and coupled nonlinear oscillators [33,59] have assumed hierarchical nesting within levels, with no overlap between subgroups within a level, a claim that does not permit group members to hold more than one role or to move between subgroups (as suggested by emergent norms theory and Figure 1A).…”