Flow topology classification of limit cycles

Abstract

Recent topological tools offer a powerful way to classify how phases of nonlinear bosonic resonators are organized. Yet, they remain incomplete. In particular, self-sustained oscillations in the form of limit cycles act as robust organizing centers in phase space that are not captured by existing fixed-point-based approaches. In this work, we extend the flow topology framework for nonlinear resonators to include limit cycles as fundamental topological elements. Using a graph-based construction, we show how periodic attractors impact the global connectivity of phase-space flows. We illustrate the approach with a minimal nonlinear Van der Pol resonator model, where limit cycles coexist with stationary points. Our results provide a unified topological description of stationary and time-periodic phases in nonlinear bosonic systems, with direct relevance to photonic, superconducting, and optomechanical platforms, and raise new questions on synchronization and the extension of flow topology to the quantum regime.