We performed numerical simulations with the Kuramoto model and experiments with oscillatory nickel electrodissolution to explore the dynamical features of the transients from random initial conditions to a fully synchronized (one-cluster) state. The numerical simulations revealed that certain networks (e.g., globally coupled or dense Erdős–Rnyi random networks) showed relatively simple behavior with monotonic increase of the Kuramoto order parameter from the random initial condition to the fully synchronized state and that the transient times exhibited a unimodal distribution. However, some modular networks with bridge elements were identified which exhibited non-monotonic variation of the order parameter with local maximum and/or minimum. In these networks, the histogram of the transients times became bimodal and the mean transient time scaled well with inverse of the magnitude of the second largest eigenvalue of the network Laplacian matrix. The non-monotonic transients increase the relative standard deviations from about 0.3 to 0.5, i.e., the transient times became more diverse. The non-monotonic transients are related to generation of phase patterns where the modules are synchronized but approximately anti-phase to each other. The predictions of the numerical simulations were demonstrated in a population of coupled oscillatory electrochemical reactions in global, modular, and irregular tree networks. The findings clarify the role of network structure in generation of complex transients that can, for example, play a role in intermittent desynchronization of the circadian clock due to external cues or in deep brain stimulations where long transients are required after a desynchronization stimulus.

我们使用 Kuramoto 模型进行了数值模拟，并使用振荡镍电溶解进行了实验，以探索从随机初始条件到完全同步（单簇）状态的瞬态的动力学特征。数值模拟表明，某些网络（例如，全局耦合或密集的 Erdős-Rnyi 随机网络）表现出相对简单的行为，仓本阶参数从随机初始条件单调增加到完全同步状态，并且瞬态时间表现出单峰分配。然而，一些具有桥接元件的模块化网络被识别出来，它们表现出具有局部最大值和/或最小值的阶次参数的非单调变化。在这些网络中，瞬态时间的直方图变为双峰，平均瞬态时间与网络拉普拉斯矩阵的第二大特征值的大小成反比。非单调瞬态将相对标准偏差从大约 0.3 增加到 0.5，即瞬态时间变得更加多样化。非单调瞬变与模块同步但彼此大致反相的相位模式的生成有关。数值模拟的预测在全局、模块化和不规则树网络中的一组耦合振荡电化学反应中得到证实。研究结果阐明了网络结构在生成复杂瞬态中的作用，例如，