Dynamics of Multifrequency Oscillator Communities

Abstract

We consider a generalization of the Kuramoto model of coupled oscillators to the situation where communities of oscillators having essentially different natural frequencies interact. General equations describing possible resonances between the communities' frequencies are derived. The simplest situation of three resonantly interacting groups is analyzed in detail. We find conditions for the mutual coupling to promote or suppress synchrony in individual populations and present examples where the interaction between communities leads to their synchrony or to a partially asynchronous state or to a chaotic dynamics of order parameters.

Intercommunity resonances in multifrequency ensembles of coupled oscillators

“…However, in two special cases equation (19) can be reduced to a simple polynomial equation with analytic solutions available. Namely, (i) for ε 2 = 0, the problem reduces to a complex quadratic equation, and (ii) for the special case Ω = δ = 0 and v = 0, equation (19) reduces to a real cubic equation. The latter case corresponds to the simplest situation when there are no phase shifts in coupling functions: α 1,2 = β = 0.…”

“…In general case solution of (19) can not be represented in an analytic form and one should use certain numerical methods to find them (a parametric representation of solutions may be possible, but we already have four auxiliary parameters, introducing another two appears not practical). However, in two special cases equation (19) can be reduced to a simple polynomial equation with analytic solutions available.…”

“…We mention here for completeness, that several nonlinear coupled populations of oscillators have been treated in refs. [19,20]. In [19] three populations of oscillators have been considered, with a resonance condition Ω 1 + Ω 2 = Ω 3 , here the coupling terms contain combinations of three phases like φ 1 + φ 2 − φ 3 .…”

“…[19,20]. In [19] three populations of oscillators have been considered, with a resonance condition Ω 1 + Ω 2 = Ω 3 , here the coupling terms contain combinations of three phases like φ 1 + φ 2 − φ 3 . In [20], a non-resonant situation has been studied, here the coupling terms are phase-independent.…”

“…A natural setup here implies existence of a certain number of oscillator subpopulations (communities), such that the frequencies inside each population are close, but differ significantly across the distinct communities [18][19][20]. This situation is inspired by theoretical and experimental results from neuroscience [21], indicating that distinct interacting brain areas exhibit different natural oscillatory rhythms.…”