Onset of synchronization in networks of second-order Kuramoto oscillators with delayed coupling: Exact results and application to phase-locked loops
We consider the inertial Kuramoto model of N 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 N→∞, 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 N, and highlight the subtleties arising from having a finite number of oscillators.
- Research Organization:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE Laboratory Directed Research and Development (LDRD) Program; Science and Engineering Research Board (SERB)
- Grant/Contract Number:
- 89233218CNA000001; TAR/2018/000023; MTR/2019/000560; 03VP06431
- OSTI ID:
- 1619881
- Alternate ID(s):
- OSTI ID: 1648082
- Report Number(s):
- LA-UR-19-25109; PPRHAI; 023183
- Journal Information:
- Physical Review Research, Journal Name: Physical Review Research Vol. 2 Journal Issue: 2; ISSN 2643-1564
- Publisher:
- American Physical Society (APS)Copyright Statement
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
GENERAL PHYSICS
Classical statistical mechanics
collective dynamics
continuum mechanics
coupled oscillators
kinetic theory
nonequilibrium statistical mechanics
synchronization
synchronization transition
continuum particle models
dynamical systems
chaos & nonlinear dynamics
dynamical mean field theory
Fokker-Planck equation
mathematical physics methods
molecular dynamics
phase oscillator models