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

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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.