Bifurcations in coupled oscillator systems: Exact results

 In the context of a paradigmatic nonlinear dynamical system of coupled oscillators with distributed natural frequencies and interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and, in the special case of a Lorentzian distribution, to the corresponding reduced dynamics on a special manifold in the space of all possible distribution functions. Besides elucidating the effects of delay on the nature of bifurcation, we show that the reduced dynamics gives an amplitude evolution of unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. The manifold thus acts as an attractor in the space of distribution functions. Such an attracting property has parallel in integrable systems as well in certain nonlinear dynamical systems, albeit with a difference: in these cases, the attracting manifold exists in the phase space of the system, while we go beyond this picture in showing the existence of such an attracting manifold in the space of distribution functions. We have demonstrated the validity of such a dynamical scenario in other systems, e.g., in a network of phase-locked loops widely used in electronic circuits, as well as in a system of coupled nonlinear oscillators with both a mean-field and a non-local interaction on a one-dimensional periodic lattice.