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 agents 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 K (+) c > 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 = K c . Although there are incoherent equilibria for whichc , there are also incoherent equilibria with a range of possiblec . How can the possible instability of incoherent states arising at K = K c < K (+) c be reconciled with the previous finding that, at large time (t → ∞), the state is always incoherent unless K > K (+) c ? We find, for a given incoherent equilibrium, that, ifc , 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 K c value exceeds that of the increased K. Thus this process, which we call 'Instability-Mediated Resetting,' leads to an increase in the effective K c with continuously increasing K, until the equilibrium has been effectively set to one for which for which K c ≈ K (+) c . Thus Instability-Mediated Resetting leads to a unique critical point of the t → ∞ time asymptotic state (K = K (+) c ) in spite of the existence of an infinity of possible pretransition incoherent states.

Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model

“…41 demonstrated that the Kuramoto model generalized to even dimensions D ≥ 4 exhibits the unusual behavior of Instability-Mediated Resetting: If the coupling strength K is increased abruptly while remaining below the critical coupling strength for the onset to coherence, the Kuramoto model displays a short burst of coherence (see Ref. 41 for further details). This is illustrated by the results shown in Fig.…”

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