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arXiv 1902.05159nlin.AOnlin.CDphysics.app-ph
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从宏观爆发观察微观跃迁：$D$维广义Kuramoto模型非相干区的不稳定性介导的重置

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

Sarthak Chandra, Edward Ott

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15 pages, 6 figures
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摘要：

我们考虑了最近引入的一个$D$维广义Kuramoto模型，其中多个$(N\gg 1)$交互主体的状态是$D$维单位向量。以前的研究表明，即使是$D$，类似于原来的仓本模型（$D=2$），存在一个连续的动力学相变，从不相干的时间渐近吸引态的相干耦合参数$K$增加通过一个临界值$K_c^{(+)}>0$。我们从非相干状态稳定性的角度来考虑这种转变，即，其中$N\to\infty$分布函数与时间无关，宏观序参量为零。与$D=2$相反，对于甚至$D>2$，存在无穷多个可能的不相干平衡，每个不相干平衡在不同的点$K=K_c$随着$K$的增加而变得不稳定。我们发现，有不相干的平衡范围内的所有$K_c$ $(K_c^{(+)}/2)\leq K_c \leq K_c^{(+)}$。在$K=K_c<K_c^{(+)}$处产生的非相干态的可能不稳定性如何与先前的发现相协调，即在大的时间内，除非$K>K_c^{(+)}$，否则状态总是非相干的？我们发现，对于一个给定的非相干平衡，即，如果$K$是迅速增加，从$K<K_c$到$K_c<K<K_c^{(+)}$，由于不稳定性，一个短的，宏观的相干性爆发观察，它最初呈指数增长，但然后达到最大值，过去它衰减回不相干。在这种衰减之后，我们观察到平衡已被重置为$K_c$值超过增加的$K$值的平衡。因此，这个过程，我们称之为“不稳定性介导的重置”，导致有效$K_c$随着$K$的不断增加而增加，直到平衡被有效地设置为一个$K_c\approx K_c^{(+)}$，导致时间渐近状态的唯一临界点。

英文摘要：

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.

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