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

We consider a recently introduced $D$-dimensional generalized Kuramoto model for many $(N\gg 1)$ interacting agents in which the agent states are $D$-dimensional unit vectors. It was previously shown that, for even $D$, 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 as the coupling parameter $K$ increases through a critical value $K_c^{(+)}>0$. We consider this transition from the point of view of the stability of an incoherent state, i.e., where the $N\to\infty$ 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 $K=K_c$. We show that there are incoherent equilibria for all $K_c$ within the range $(K_c^{(+)}/2)\leq K_c \leq K_c^{(+)}$. 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, 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, which initially grows exponentially, but then reaches a maximum, past which it decays back into incoherence. After this decay, we observe that the equilibrium has been reset to one 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 $K_c\approx K_c^{(+)}$, leading to a unique critical point of the time asymptotic state.