Observing microscopic transitions from macroscopic bursts: Instability-mediated resetting in the incoherent regime of the D-dimensional generalized Kuramoto model

Abstract

This paper considers a recently introduced D-dimensional generalized Kuramoto model for many (N≫1) interacting agents, in which the agent states are D-dimensional unit vectors. It was previously shown that, for even (but not odd) D, similar to the original Kuramoto model (D=2), there exists a continuous dynamical phase transition from incoherence to coherence of the time asymptotic attracting state (time t→∞) as the coupling parameter K increases through a critical value which we denote Kc(+)>0. We consider this transition from the point of view of the stability of an incoherent state, where an incoherent state is defined as one for which the N→∞ distribution function is time-independent and the macroscopic order parameter is zero. In contrast with D=2, for even D>2, there is an infinity of possible incoherent equilibria, each of which becomes unstable with increasing K at a different point K=Kc. Although there are incoherent equilibria for which Kc=Kc(+), there are also incoherent equilibria with a range of possible Kc values below Kc(+), (Kc(+)/2)≤Kc<Kc(+). How can the possible instability of incoherent states arising at K=Kc<Kc(+) be reconciled with the previous finding that, at large time (t→∞), the state is always incoherent unless K>Kc(+)? We find, for a given incoherent equilibrium, that, if K is rapidly increased from K<Kc to Kc<K<Kc(+), due to the instability, a short, macroscopic burst of coherence is observed, in which the coherence initially grows exponentially, but then reaches a maximum, past which it decays back into incoherence. Furthermore, after this decay, we observe that the equilibrium has been reset to a new equilibrium whose Kc value exceeds that of the increased K. Thus, this process, which we call “Instability-Mediated Resetting,” leads to an increase in the effective Kc with continuously increasing K, until the equilibrium has been effectively set to one for which Kc≈Kc(+). Thus, instability-mediated resetting leads to a unique critical point of the t→∞ time asymptotic state (K=Kc(+)) in spite of the existence of an infinity of possible pretransition incoherent states.