Synchronization in populations of globally coupled oscillators with inertial effects

Abstract

A model for synchronization of globally coupled phase oscillators including "inertial" effects is analyzed. In such a model, both oscillator frequencies and phases evolve in time. Stationary solutions include incoherent (unsynchronized) and synchronized states of the oscillator population. Assuming a Lorentzian distribution of oscillator natural frequencies, g(Ω), both larger inertia or larger frequency spread stabilize the incoherent solution, thereby making harder to synchronize the population. In the limiting case g(Ω) = δ(Ω), the critical coupling becomes independent of inertia. A richer phenomenology is found for bimodal distributions. For instance, inertial effects may destabilize incoherence, giving rise to bifurcating synchronized standing wave states. Inertia tends to harden the bifurcation from incoherence to synchronized states: at zero inertia, this bifurcation is supercritical (soft), but it tends to become subcritical (hard) as inertia increases. Nonlinear stability is investigated in the limit of high natural frequencies.

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

“…[29,30,31,32,33,20]. Inclusion of inertia elevates the first-order Kuramoto dynamics to one that is second order in time, while noise accounts for temporal fluctuations of the natural frequencies.…”

“…Similar to what was done for the Kuramoto model, we define a single-oscillator density f (θ, v, ω, t) that gives at time t and for each ω the fraction of oscillators that have angle θ and angular velocity v. The density f is 2π-periodic in θ, obeys the normalization π −π dθ +∞ −∞ dv f (θ, v, ω, t) = 1 ∀ ω, t, and has a time evolution given by the so-called Kramers equation [28,32,33,20] …”

“…For σ = 0, the θ-independent solution characterizing the incoherent phase, for which r st = 0, is given by [32]:…”

“…(48), and keeping terms to linear order in δf . The solution of the linearized equation yields the following equation that λ has to satisfy [32]:…”

“…In the generalized dynamics, a dynamical variable in addition to the angle θ j , namely, angular velocity v j , is assigned to each oscillator, so that the equations of motion are [30,31,32]:…”