Investigating the dynamics of a network of oscillatory systems is a timely and urgent topic. Phase synchronization has proven paradigmatic to study emergent collective behavior within a network. Defining the phase dynamics, however, is not a trivial task. The literature provides an arsenal of solutions, but results are scattered and their formulation is far from standardized. Here, we present, in a unified language, a catalogue of popular techniques for deriving the phase dynamics of coupled oscillators. Traditionally, approaches to phase reduction address the (weakly) perturbed dynamics of an oscillator. They fall into three classes. (i) Many phase reduction techniques start off with a Hopf normal form description, thereby providing mathematical rigor. There, the caveat is to first derive the proper normal form. We explicate several ways to do that, both analytically and (semi-)numerically. (ii) Other analytic techniques capitalize on time scale separation and/or averaging over cyclic variables. While appealing for their more intuitive implementation, they often lack accuracy. (iii) Direct numerical approaches help to identify oscillatory behavior but may limit an overarching view how the reduced phase dynamics depends on model parameters. After illustrating and reviewing the necessary mathematical details for single oscillators, we turn to networks of coupled oscillators as the central issue of this report. We show in detail how the concepts of phase reduction for single oscillators can be extended and applied to oscillator networks. Again, we distinguish between numerical and analytic phase reduction techniques. As the latter dwell on a network normal form, we also discuss associated reduction methods. To illustrate benefits and pitfalls of the different phase reduction techniques, we apply them point-by-point to two classic examples: networks of Brusselators and a more elaborate model of coupled Wilson–Cowan oscillators. The reduction of complex oscillatory systems is crucial for numerical analyses but more so for analytical estimates and model prediction. The most common reduction is towards phase oscillator networks that have proven successful in describing not only the transition between incoherence and global synchronization, but also in predicting the existence of less trivial network states. Many of these predictions have been confirmed in experiments. As we show, however, the phase dynamics depends to large extent on the employed phase reduction technique. In view of current and future trends, we also provide an overview of various methods for augmented phase reduction as well as for phase–amplitude reduction. Weindicate how these techniques can be extended to oscillator networks and, hence, may allow for an improved derivation of the phase dynamics of coupled oscillators.

中文翻译：

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耦合振荡器的网络动力学和减相技术

研究振荡系统网络的动力学是一个及时而紧迫的话题。相位同步已被证明是研究网络内突发集体行为的范例。然而，定义相位动力学并不是一项微不足道的任务。文献提供了一系列解决方案，但结果是零散的，而且它们的表述远非标准化。在这里，我们以统一的语言展示了用于推导耦合振荡器相位动力学的流行技术目录。传统上，减少相位的方法解决振荡器的（弱）扰动动态。他们分为三类。(i) 许多减相技术从 Hopf 范式描述开始，从而提供数学严谨性。在那里，需要注意的是首先导出正确的范式。我们解释了几种方法来做到这一点，包括分析和（半）数字。(ii) 其他分析技术利用时间尺度分离和/或循环变量的平均。虽然它们更直观的实现很有吸引力，但它们往往缺乏准确性。(iii) 直接数值方法有助于识别振荡行为，但可能会限制减少的相位动力学如何取决于模型参数的总体观点。在说明和审查了单个振荡器的必要数学细节后，我们将耦合振荡器网络作为本报告的中心问题。我们详细展示了如何将单个振荡器的减相概念扩展并应用于振荡器网络。再次，我们区分数值和解析相位缩减技术。由于后者涉及网络范式，我们还讨论了相关的约简方法。为了说明不同减相技术的优点和缺点，我们将它们逐点应用于两个经典示例：布鲁塞尔网络和更精细的耦合威尔逊-考恩振荡器模型。复杂振荡系统的减少对于数值分析至关重要，但对于分析估计和模型预测更是如此。最常见的减少是针对相位振荡器网络，它已被证明不仅成功地描述了不相干和全局同步之间的转换，而且还预测了不太重要的网络状态的存在。许多这些预测已在实验中得到证实。然而，正如我们所展示的，相动力学在很大程度上取决于所采用的相减少技术。鉴于当前和未来的趋势，我们还概述了用于增强相位降低以及相位幅度降低的各种方法。我们指出如何将这些技术扩展到振荡器网络，从而可以改进耦合振荡器的相位动力学推导。