SIAM Review, Volume 66, Issue 4, Page 619-679, November 2024.
There is enormous interest---both mathematically and in diverse applications---in understanding the dynamics of coupled-oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled-oscillator networks. We discuss phase reductions, phase--amplitude reductions, and the master stability function for smooth dynamical systems. We then focus, in particular, on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use saltation operators, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds.

中文翻译：

---

振荡网络：来自分段线性建模的见解


SIAM 评论，第 66 卷，第 4 期，第 619-679 页，2024 年 11 月。

人们---无论是在数学上还是在各种应用中---都对理解耦合振荡器网络的动力学产生了巨大的兴趣。这种网络的现实动机来自对大脑、心脏、生态学等的研究。通常用反映网络组件的连通性和非线性动力学的网络活动的复杂模式来描述这些系统中丰富的涌现行为。这种行为通常是围绕锁相周期性状态及其不稳定性组织的。然而，众所周知，非线性系统（即使是在低维）中周期性轨道的显式计算是很困难的，因此网络级洞察通常需要对一些潜在的周期性分量进行数值构造。在本文中，我们回顾了研究耦合振荡器网络的强大技术。我们讨论了相位减少、相位幅度减少和平滑动力学系统的主稳定函数。然后，我们特别关注这些方法的增强，以分析分段线性系统，为此，人们可以很容易地构建周期性轨道。这为网络行为提供了有用的见解，但代价是需要研究非光滑动态系统。当关注系统的相互作用单元（即在节点级别）时，对非光滑系统的研究已经发展得很好，我们详细介绍了如何使用盐化算子，它可以处理扰动通过切换流形的传播，以理解网络级的动力学和分叉。 我们说明了网络科学和非光滑动力学系统的工具和技术的这种融合，以及应用于神经系统、心脏系统、机电振荡器网络以及牛群合作。