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  • Open Access

Flow topology classification of limit cycles

Thomas Mutschler, Greta Villa, and Oded Zilberberg*

  • *Contact author: oded.zilberberg@uni-konstanz.de
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Phys. Rev. Research 8, 033213 – Published 20 August, 2026

DOI: https://doi.org/10.1103/fwpm-41bj

Abstract

Recent advancements in topological analysis offer a powerful way to classify how phases of nonlinear bosonic resonators organize. These tools are rooted in the Morse-Smale flow topology framework, where a graph-based construction describes how steady-state solutions are arranged in phase space. This construction remains incomplete as self-sustained oscillations in the form of limit cycles can impact the global connectivity of phase-space flows and the corresponding Morse-Smale topological classification. In this work, we extend the bosonic classification scheme to include limit cycles as fundamental topological elements. We illustrate the approach with a minimal nonlinear Van der Pol resonator model, where limit cycles coexist with steady-state solutions. 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. Our findings raise fundamental questions regarding synchronization and the extension of flow topology to the quantum regime.

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Phys. Rev. Research 8, 033213– Published 20 August, 2026
  • Received 3 January 2026
  • Accepted 1 July 2026
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