2019/12/31 by Evelyn Sander, E. Sander, J. D. Meiss +1 · 1 voice
Mathematics · Physics and Astronomy · #Algorithm #Artificial intelligence #Chaos control and synchronization #Chaotic #Computer science #Geometry #Homogeneous space #Invariant (physics) #Kolmogorov–Arnold–Moser theorem #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Quantum chaos and dynamical systems #Rotation (mathematics) #Rotation number #Rotational invariance #Standard map #Torus #math.DS #msc:37E40 #msc:37E45 #msc:37J10 #nlin.CD
paper · pdf · doi:10.1016/j.physd.2020.132569
published as Physica D 411 132569 (2020)
arxiv created 2019/12/31 · arxiv published 2019/12/31 · openalex created_date 2020/01/10 · openalex publication_date 2020/05/21 · arxiv updated 2020/06/02 · openalex updated_date 2026/08/05
Rotational invariant circles of area-preserving maps are an important and well-studied example of KAM tori. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. The accurate numerical confirmation of these conjectures relies on the map having a time reversal symmetry, and these methods cannot be applied to more general maps. In this paper, we develop a method based on a weighted Birkhoff average for identifying chaotic orbits, island chains, and rotational invariant circles that do not rely on these symmetries. We use Chirikov's standard map as our test case, and also demonstrate that our methods apply to three other, well-studied cases.