Observing Microscopic Transitions from Macroscopic Bursts: Instability-Mediated Resetting in the Incoherent Regime of the D-Dimensional Generalized Kuramoto Model

Chaos 29, 033124 (2019)https://ireap.umd.edu/10.1063/1.50849652019Journal ArticleComplex and Emergent Systems

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 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 which K_c = K⁽⁺⁾_c ⁠, there are also incoherent equilibria with a range of possible K_c values below K⁽⁺⁾_c⁠, (K⁽⁺⁾_c/2) ≤ K_c < K⁽⁺⁾_c⁠. 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, if K is rapidly increased from K < K_c to K_c < K < K⁽⁺⁾_c ⁠, 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 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.