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

同步是非线性网络动力学的一种普遍现象，在自然和人造系统中起着至关重要的作用。耦合相位振子的仓本模型构成了同步过程的一个范例模型，在物理学、生物学和工程学中有着广泛的应用。在连续极限下，关于仓本模型的分析陈述导致了同步跃迁的简明理论。然而，到目前为止，有限N系统在很大程度上回避了分析访问，限制了全面的理解。在这里，我们建议调查的集体动力学的仓本模型与传统的真实的状态变量解析继续复杂。在过去，解析延拓的复杂化一再催化我们理解跨系统类的非线性动力学的重大进展。例如，它使相变、分形结构和PT对称量子力学的分析理论成为可能。该项目的主要方向是，首先，发现和解释复杂耦合和网络化动力学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模型中的有序现象，这反过来又是许多应用的基础。我们的初步分析揭示了存在的广义形式复杂的锁定状态，也坚持弱耦合。此外，数值结果表明，这些复杂的锁定状态的稳定性告诉我们的频率锁定的子群体的存在，在真实的模型，与虚部告诉我们哪些单位属于这些子群体。在拟议的项目中，我们计划解决三个主要类别的问题，有助于巩固我们对这种广义锁定形式的关键总体原则的理解，重点是有限N系统。首先，在最初的真实的模型中，是什么机制将复杂锁定态的稳定性与频率锁定子种群的存在或不存在联系起来？第二，我们如何评估一个适当的序参数，以了解复杂的仓本网络的协调程度？第三，在复杂网络中，哪些新的集体动力学类型也超越了基本的仓本模型？它们背后的机制是什么？一个成功的项目不仅将复杂的锁定状态连接到传统锁定和传统解锁状态的原始，实变量仓本有限N模型，从而链接到几个应用程序。它还将揭示新的机制，并扩大我们的角度来理解协调现象，通过解析延拓方法耦合多维动力系统一般。

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.