Synchronization and Collective Nonlinear Dynamics in Complexified Oscillator Networks — SynCON —

同步是一种普遍存在的非线性网络动力学现象，在自然系统和人为系统中起着至关重要的作用。耦合相位振荡器的Kuramoto模型是同步过程的一个典型模型，在物理、生物和工程中有着广泛的应用。在连续体极限下，关于Kuramoto模型的解析性陈述导致了同步跃迁的简明理论。然而，有限n系统迄今为止在很大程度上回避了分析访问，限制了全面的理解。在这里，我们建议研究Kuramoto模型的集体动力学，其传统的真实状态变量分析继续是复杂的。在过去，通过解析延拓的复化已经多次催化了我们对系统类间非线性动力学的理解的重大进展。例如，它使相变、分形结构和PT对称量子力学的解析理论成为可能。提出的项目的主要方向是，首先，揭示和解释复杂耦合和网络动力单元的基本类中的新现象，重点是Kuramoto模型，其次，利用这些见解来具体理解有限n实变量Kuramoto模型中的排序现象，这反过来又是许多应用的基础。我们的初步分析揭示了广义形式的复杂锁定状态的存在，这种状态在弱耦合下也会持续存在。此外，数值结果表明，这些复杂锁定状态的稳定性告诉我们在真实模型中存在频率锁定子种群，虚部告诉我们哪些单元属于这些子种群。在提议的项目中，我们计划解决三个主要类别的问题，这些问题有助于并巩固我们对这种广义锁形式的关键总体原则的理解，重点是有限n系统。首先，在原始的真实模型中，哪些机制将复杂锁定状态的稳定性与频率锁定子种群的存在或不存在联系起来？其次，我们如何评估一个合适的顺序参数来了解复杂Kuramoto网络的协调程度？第三，在复杂网络中出现了哪些新的集体动力学类型，这些类型也超出了基本的Kuramoto模型？它们背后的机制是什么？一个成功的项目不仅可以将复杂的锁定状态与有限n的原始实变量Kuramoto模型中的传统锁定状态和传统解锁状态连接起来，并且可以连接到多个应用程序。它还将揭示新的机制，并扩展我们通过解析延拓方法理解耦合多维动力系统的协调现象的视角。

Synchronization constitutes a ubiquitous phenomenon of nonlinear network dynamics and plays an essential role across natural and human-made systems. The Kuramoto model of coupled phase-oscillators constitutes a paradigmatic model for synchronization processes, with many applications in physics, biology and engineering. In the continuum limit, analytical statements about the Kuramoto model have led to a concise theory of the synchronization transition. However, finite-N systems have so far largely evaded analytic access, limiting a comprehensive understanding. Here we propose to investigate the collective dynamics of the Kuramoto model with their traditionally real state variables analytically continued to be complex. In the past, complexification by analytic continuation has repeatedly been catalyzing major progress in our understanding the nonlinear dynamics across system classes. For instance, it has enabled an analytic theory of phase transitions, of fractal structures and PT symmetric quantum mechanics. The main direction of the proposed project is to, first, uncover and explain novel phenomena in a fundamental class of complexified coupled and networked dynamical units, with a focus on the Kuramoto model, and second, to exploit those insights to specifically understand ordering phenomena in the finite-N real-variable Kuramoto model that in turn underlies many applications. Our preliminary analysis has revealed the existence of generalized forms complex locked states that persist also for weak coupling. Moreover, numerical results indicate that the stability of those complex locked states informs us about the existence of frequency-locked sub-populations in the real model, with the imaginary parts informing us about which units belong to those subpopulations. In the proposed project, we plan to address three main classes of questions contributing to and consolidating our understanding of key overarching principles underlying such generalized forms of locking, with a focus on finite-N systems. First, which mechanisms link the stability of the complex locked states to the existence or non-existence of frequency-locked sub-populations in the original, real model? Second, how we evaluate an appropriate order parameter to learn about the degree of coordination in complexified Kuramoto networks? Third, which new types of collective dynamics emerge in classes of complexified networks also beyond the basic Kuramoto model? What are the mechanisms underlying them? A successful project would not only connect the complex locked states to both traditionally locked and traditionally unlocked states in the original, real-variable Kuramoto model for finite-N, and thereby link to several applications. It would also reveal novel mechanisms and expand our perspective on understanding coordination phenomena via analytic continuation methods for coupled multi-dimensional dynamical systems in general.