Nonstandard transitions in the Kuramoto model: a role of asymmetry in natural frequency distributions

Abstract

Abstract. We study transitions in the Kuramoto model by shedding light on asymmetry in the natural frequency distribution, which has been assumed to be symmetric in many previous studies. The asymmetry brings two nonstandard bifurcation diagrams, with the aid of bimodality. The first diagram consists of stationary states, and has the standard continuous synchronization transition and a subsequent discontinuous transition as the coupling strength increases. Such a bifurcation diagram has been also reported in a variant model, which breaks the odd symmetry of the coupling function by introducing the phase lag. The second diagram includes the oscillatory state emerging from the partially synchronized state and followed by a discontinuous transition. This diagram is firstly revealed in this study. The two bifurcation diagrams are obtained by employing the Ott-Antonsen ansatz, and are verified by direct N -body simulations. We conclude that the asymmetry in distribution, with the bimodality, plays a similar role to the phase lag, and diversifies the transitions.

Dynamics in the Sakaguchi-Kuramoto model with bimodal frequency distribution

“…Positive and negative ω 0 may exert different impacts on the model dynamics due to the asymmetrical bimodal frequency distribution. Incoherent states change stability with changing parameters at hopf bifurcation or pitchfork bifurcation [25,26,29,31,35]. We can get the bifurcation curves in Fig 5 from Eq (15), in which the more general conditions are considered analytically.…”

“…Above K c , the incoherent state yields to a stationary partial synchronous state. For asymmetrical unimodal g(ω), the partial synchronous states are always time-dependent [26]. When g(ω) becomes a bimodal one, increasing coupling strength always first leads to a standing wave state, in which two synchronous clusters of oscillators oscillate at opposite mean frequencies and, then, to traveling wave states, in which synchronous oscillators rotate at the same frequency [27].…”

Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions

“…When exists, the synchronized uniformly twisted stationary state is represented as u syn st (x) = a(q)e iqx , with a(q) given by Eqs. (54) and (55).…”

“…Using r st = a(q 0 = 0)δ q 0 ,0 , Z st (x) = a(q 0 )e −iq 0 x G(q 0 ), with a(q 0 ) and a(q 0 = 0) given respectively by Eqs. (54) and (55), and writing u(x, t) as…”

Classification of bifurcation diagrams in coupled phase-oscillator models with asymmetric natural frequency distributions

“…A significant part of previous studies assumes symmetry of natural frequency distributions, and bifurcation is found as the synchronization transition referring to the transition from the nonsynchronized state to partially synchronized states. Interestingly, symmetry breaking brings new types of bifurcations that emerge not from the nonsynchronized state but from partially synchronized states and are followed by a discontinuous jump or oscillation of the order parameter [14]. The new types of bifurcations have been observed by performing numerical simulations, and this paper aims to propose a theoretical explanation of the new types of bifurcations.…”