标题： 从网络Kuramoto模型的不一致到同步 显示英文标题

标题： From incoherence to synchronicity in the network Kuramoto model

摘要： 我们研究了一般网络上耦合相位振荡器的Kuramoto模型的同步特性。 在这里，我们区分了该系统自我同步的能力与这种行为的稳定性。 虽然自我同步是真正非微扰动力学的结果，但动力系统中的稳定性通常可以通过关于固定点的波动来分析，这里取同步解作为固定点。 我们通过图拉普拉斯矩阵的模式来研究这个问题，从而可以轻易地证明同步固定点的绝对Lyapunov稳定性。 在波动的下一级中可以看到稳定性偏离，此时动力学方程类似于物种种群模型的动力学方程，如逻辑斯蒂方程和Lotka-Volterra方程。 利用这些系统的的方法，我们解析地推导出新的临界耦合，表示偏离经典稳定性的信号。 我们在某些情况下观察到介于无序和同步之间的中间行为区域，其中表现出系统范围内的周期性行为。 我们根据模拟结果讨论这些结果。 显示英文摘要

摘要： We study the synchronisation properties of the Kuramoto model of coupled phase oscillators on a general network. Here we distinguish the ability of such a system to self-synchronise from the stability of this behaviour. While self-synchronisation is a consequence of genuine non-perturbative dynamics, the stability in dynamical systems is usually accessible by fluctuations about a fixed point, here taken to be the synchronised solution. We examine this problem in terms of modes of the graph Laplacian, by which the absolute Lyapunov stability of the synchronised fixed point is readily demonstrated. Departures from stability are seen to arise at the next order in fluctuations where the dynamical equations resemble those for species population models, the logistic and Lotka-Volterra equations. Methods from these systems are exploited to analytically derive new critical couplings signalling deviation from classical stability. We observe in some cases an intermediate regime of behaviour, between incoherence and synchronisation, where system wide periodic behaviours are exhibited. We discuss these results in light of simulations.