2012/09/13 by Livia Corsi, Roberto Feola, Guido Gentile · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Codimension #Diophantine equation #Hamiltonian (control theory) #Hamiltonian system #Homogeneous space #Integrable system #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Rotation number #Torus #math-ph #math.CA #math.DS #math.MP #msc:37J20 #msc:37J40 #msc:58F27
paper · pdf · doi:10.1007/s10955-012-0682-8
published as Journal of Statistical Physics 150 (2013), no. 1, 156-180 · 27 pages
arxiv created 2012/09/13 · openalex publication_date 2013/01/01 · arxiv updated 2015/06/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.