A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

• DOI: 10.1007/s00220-021-04077-z
• Publication Date: 2021-04-08T02:03:27Z
• Authors: Maximilian Engel; Christian Kuehn
• Keywords: Isochron dating; Eigenfunction; Limit cycle; Orbit (dynamics)

Abstract For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point as cross-section with fixed return time under flow. Equivalently, isochrons can be characterized stable manifolds foliating neighborhoods limit cycle or level sets map. In recent years, there has been lively discussion in mathematical physics community on how to define stochastic oscillations, i.e. cycles heteroclinic exposed noise. The main concerned approach finding sections equal expected times versus idea considering eigenfunctions backward Kolmogorov operator. We discuss problem framework random systems and introduce new rigorous definition solutions noise-dependent period. This allows us establish version maps whose coincide manifolds. Finally, we links between interpretation via averaged quantities.