Low-dimensional and Group-theoretic Reduction Techniques for Coupled Oscillator Networks

耦合振荡器网络在自然界和科学中比比皆是；心脏起搏器细胞、化学振荡、超导电路和萤火虫群都提供了振荡系统相互交流或相互影响的例子，从而协调整个网络的集体行为。这些系统通常由大量单独的振荡器组成，因此它们的行为可能非常复杂且难以分析。但这些复杂网络的集体行为往往出奇地简单；例如，振荡器可能同步，就像心脏起搏器网络或某些种类的萤火虫同步以一致闪光一样。振荡器网络的其他简单集体行为包括锁相配置，其中所有振荡器收敛到具有相同的周期，或张开状态，这是振子在相位上均匀分布的配置。理解这些简单的集体行为背后的数学原理是应用动力系统中一个非常重要的研究问题。本研究的一个持久主题是，耦合振荡器网络的集体动力学通常可以简化为低维系统的动力学，通常是通过系统中隐藏的不变性或对称性的存在。重要的例子是由相同的Kuramoto振子组成的网络，这些振子是由三角函数控制的简单振子。这种网络在被称为莫比乌斯变换的三维对称组下是不变的，莫比乌斯变换在共形双曲几何中起着关键作用。这一不变性已被用来解释Kuramoto网络的稳定性性质，并对相同Kuramoto振子网络的稳定长期动力学进行了完全分类。振荡器网络的低维化也可以通过其他技术来实现，例如复解析延拓和剩余方法，这些方法也被用于经典Kuramoto模型的化简。该研究项目将建立在这些强大的方法上，以扩展和进一步阐明我们对振荡器网络中集体行为的理解。振子网络的集体行为研究项目分为四个部分。项目1将使用Mobius群、Riccati方程和线性代数技术来研究多种群Kuramoto振荡器网络是否支持“嵌合体状态”的一般问题，在这种状态下，一部分种群同步，而其余部分稳定地保持异步。项目2将使用分析技术，包括Fokker-Planck方法和无限维光谱分析，研究更一般类型的振荡器网络（包括集成和火灾网络）中同步和展态的稳定性，目的是更好地理解Kuramoto振荡器的特殊性质。项目3将应用ot - antonsen ansatz的复杂分析技术来研究随机钉住耦合自旋的模型，并了解其低维行为的来源。最后，项目4将在实验环境下研究Kuramoto网络的设计，以实现理论分类中预测的所有可能的长期稳定动态。特别是，研究人员已经确定了相同Kuramoto网络有四种可能的稳定吸引子类型：完全同步的不动点和极限环，以及除一个振子外所有振子都同步的不动点和极限环。该项目旨在在实验环境中观察这些稳定的“（n - 1,1）”状态，并探索超导阵列中这种状态对量子计算中相干性的发展和稳定性的影响。这些项目为本科生和研究生水平的跨学科参与提供了机会，范围从数值和实验模拟到理论分析调查。本研究由数学系通过应用数学项目和物理系通过生命系统物理项目共同资助。

Networks of coupled oscillators abound in nature and science; heart pacemaker cells, chemical oscillations, superconducting circuits, and swarms of fireflies all provide examples of systems of oscillators that communicate with or influence each other, and thereby coordinate collective behavior of the network as a whole. These systems typically consist of large numbers of individual oscillators, so their behavior can be extremely complicated and difficult to analyze. But often the collective behavior of these complex networks is surprisingly simple; for example, the oscillators may synchronize, as happens in the heart pacemaker network or for certain species of fireflies that synchronize so as to flash in unison. Other simple collective behaviors of oscillator networks include phase-locked configurations, in which all the oscillators converge to have the same period, or splay states, which are configurations in which the oscillators are uniformly distributed in phase. Understanding the mathematical principles that underlie or facilitate such simple collective behaviors is a research problem of central importance in applied dynamical systems. A persistent theme in this study is that the collective dynamics of coupled oscillator networks can often be reduced to that of low-dimensional systems, usually via the presence of hidden invariances or symmetries in the system. Important examples are networks consisting of identical Kuramoto oscillators, which are simple oscillators governed by trigonometric functions. Such networks are invariant under a three-dimensional group of symmetries called Mobius transformations, which play a key role in conformal hyperbolic geometry. This invariance has been used to explain the stability properties of Kuramoto networks and to completely classify the stable long-term dynamics of networks of identical Kuramoto oscillators. Low-dimensional reduction of oscillator networks can also be achieved through other techniques, such as complex-analytic continuation and residue methods, which have also been employed in reducing the classic Kuramoto model. This research project will build on such powerful methods to extend and further clarify our understanding of collective behavior in oscillator networks.This research project on collective behavior in oscillator networks has four parts. Project 1 will use Mobius group, Riccati equation, and linear algebra techniques to study the general question of whether a multi-population Kuramoto oscillator network supports "chimera states," in which a portion of the population synchronizes while the remainder is stably asynchronous. Project 2 will use analytic techniques including Fokker-Planck methods and infinite-dimensional spectral analysis to study the stability of synchronized and splay states in more general types of oscillator networks, including integrate-and-fire networks, with the goal of better understanding the exceptional nature of Kuramoto oscillators. Project 3 will apply the complex-analytic techniques of the Ott-Antonsen ansatz to study a model of coupled spins with random pinning and to understand the source of its low-dimensional behavior. Finally, project 4 will investigate the design of Kuramoto networks in an experimental setting, to realize all the possible long-term stable dynamics predicted in the theoretical classification. In particular, the investigators have established that there are four possible types of stable attractors for identical Kuramoto networks: fully synchronized fixed points and limit cycles, and fixed points and limit cycles in which all but one of the oscillators are synchronized. This project aims to observe these stable "(N-1, 1)" states in an experimental setting, as well as to explore the ramifications of such states in superconducting arrays for the development and stability of coherence in quantum computing. These projects afford opportunities for interdisciplinary participation at the undergraduate and graduate level, ranging from numerical and experimental simulations to theoretical, analytic investigations.This research is jointly funded by the Division of Mathematics through the Applied Mathematics program and the Division of Physics through the Physics of Living Systems program.