2013/06/04 by Amadeu Delshams, Delshams, Amadeu, Marina Gonchenko +4 · 1 citation
Mathematics · Physics and Astronomy · #37J40 #70H08 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #math.DS #msc:37J40 #msc:70H08
paper · pdf · doi:10.48550/arxiv.1306.0728
arxiv created 2013/06/04 · openalex publication_date 2013/06/04 · arxiv updated 2013/06/05 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems. We consider 2-dimensional tori with a frequency vector ω=(1,Ω) where Ω is a quadratic irrational number, or 3-dimensional tori with a frequency vector ω=(1,Ω,Ω2) where Ω is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, showing that such estimates depend strongly on the arithmetic properties of the frequencies. Inthe quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Ω is the so-called cubic golden number (the real root of x3+x-1=0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.