Onset of synchronization in networks of second-order Kuramoto oscillators with delayed coupling: Exact results and application to phase-locked loops
arXiv:1906.02643 · doi:10.1103/PhysRevResearch.2.023183
Abstract
We consider the inertial Kuramoto model of globally coupled oscillators characterized by both their phase and angular velocity, in which there is a time delay in the interaction between the oscillators. Besides the academic interest, we show that the model can be related to a network of phase-locked loops widely used in electronic circuits for generating a stable frequency at multiples of an input frequency. We study the model for a generic choice of the natural frequency distribution of the oscillators, to elucidate how a synchronized phase bifurcates from an incoherent phase as the coupling constant between the oscillators is tuned. We show that in contrast to the case with no delay, here the system in the stationary state may exhibit either a subcritical or a supercritical bifurcation between a synchronized and an incoherent phase, which is dictated by the value of the delay present in the interaction and the precise value of inertia of the oscillators. Our theoretical analysis, performed in the limit , is based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to the kinetic equation satisfied by the single-oscillator distribution function. We check our results by performing direct numerical integration of the dynamics for large , and highlight the subtleties arising from having a finite number of oscillators.
arXiv comment15 pages, 4 figures; v2: 16 pages, 5 figures, published version
References in corpus (13)
- The Kuramoto model in complex networks
- Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
- Time Delay in the Kuramoto Model of Coupled Oscillators
- On Landau damping
- Kuramoto model of synchronization: Equilibrium and nonequilibrium aspects
- Hysteretic transitions in the Kuramoto model with inertia
- Scaling and singularities in the entrainment of globally-coupled oscillators
- Entrainment transition in populations of random frequency oscillators
- Synchronization in populations of globally coupled oscillators with inertial effects
- Bifurcations and singularities for coupled oscillators with inertia and frustration
- Amplitude Equations for Electrostatic Waves: universal singular behavior in the limit of weak instability
- Spontaneous synchronization and nonequilibrium statistical mechanics of coupled phase oscillators
- Bifurcations in the time-delayed Kuramoto model of coupled oscillators: Exact results