发表机构
School of Mathematical Sciences, Fudan University, Shanghai 200433, China; Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China; Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom; Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom; Department of Psychiatry, University of Cambridge, Cambridge CB2 1TN, United Kingdom（复旦大学数学科学学院; 复旦大学智能复杂系统研究所; 伦敦帝国学院数学系; 阿伯丁大学复杂系统与数学生物学研究所; 剑桥大学精神病学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究非线性高维振荡中相位和相位动力学识别问题，基于托勒密等距点建立通用动态时钟原理，用机器学习框架证明其存在并构建相关动力学，通过四个发现展示价值，为振荡系统研究提供新途径。
AI 中文摘要
振荡动力学在非线性系统中普遍存在，但在非线性高维振荡中识别具有物理可解释性的相位和相位动力学仍是一个未解决的核心问题。本文建立了通用动态时钟原理，受托勒密等距点启发，通过面积均匀性原理形式化，将任意维度和几何形状的振荡等效表示为通过等距点诱导的非线性观察坐标的匀速旋转。利用机器学习框架，证明了一类广泛振荡动力学中等距点的存在，并构建了相关动态时钟和相位动力学。通过四个发现展示了其在揭示新物理规则和现象方面的价值，包括大肠杆菌群体中的集体振荡遵循超线性缩放定律、工程遗传电路对基因表达和环境条件变化的响应机制、贝里几何相位的经典力学对应物自然出现以及最优等距点非均匀性为临界转变提供几何预警信号并预测临界参数。动态时钟提供了可直接从数据构建的操作和系统无关的相位动力学，实现了对振荡系统的分类、比较和控制，为理解特定动态机制如何支持网络系统中不同功能行为提供了新途径。
英文摘要
Oscillatory dynamics arise ubiquitously in nonlinear systems, yet identifying a physically interpretable phase and phase dynamics in nonlinear, high-dimensional oscillations remains a central unresolved problem. Here we establish the principle of a universal dynamical clock, a physical perspective in which oscillations of arbitrary dimensionality and geometry are equivalently represented as uniform rotation through an equant-induced nonlinear viewing coordinate, inspired by Ptolemy's equant and formalised through an areal-uniformity principle reminiscent of Kepler's second law. Using a machine-learning framework, we demonstrate the existence of such an equant for a broad class of oscillatory dynamics and construct the associated dynamical clock and phase dynamics under additive forces, including noise, periodic perturbations, and coupling. Its value in uncovering new physical rules and phenomena is demonstrated by four findings: (i) collective oscillations in Escherichia coli populations obey a previously unexplained superlinear scaling law, resolving a long-standing open problem posed in 2004; (ii) the response mechanisms of engineered genetic circuits to changes in gene expression and environmental conditions; (iii) a classical-mechanics counterpart of the Berry geometric phase emerges naturally from the phase of the dynamical clock; and (iv) optimal equant non-uniformity provides a geometric early-warning signal for critical transitions and enables prediction of critical parameters. By providing operational and system-agnostic phase dynamics that can be constructed directly from data, the dynamical clock enables principled classification, comparison, and control of oscillatory systems, and offers a new route to understanding how specific dynamical regimes support distinct functional behaviours in networked systems.
Comments56 pages, 12 figures, 3 tables
