2015/07/27 by Amadeu Delshams, Delshams, Amadeu, Marina Gonchenko +3 · 1 citation
Mathematics · Physics and Astronomy · #Bifurcation #Combinatorics #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry #Hamiltonian system #Homoclinic orbit #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Omega #Physics #Pure mathematics #Quadratic equation #Quantum chaos and dynamical systems #Quantum mechanics #Stochastic processes and statistical mechanics #Torus #Transversality #Upper and lower bounds #math.DS
paper · pdf · doi:10.48550/arxiv.1507.07397
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2015/07/27 · arxiv created 2015/12/15 · arxiv updated 2015/12/16 · openalex created_date 2022/10/04 · openalex updated_date 2026/08/05
The splitting of invariant manifolds of whiskered (hyperbolic) tori with two\nfrequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is\ngiven by a pendulum, is studied. We consider a torus with a fast frequency\nvector \ω/\√\ε, with \ω=(1,\Ω) where the frequency\nratio \Ω is a quadratic irrational number. Applying the\nPoincar 'e-Melnikov method, we carry out a careful study of the dominant\nharmonics of the Melnikov potential. This allows us to provide an asymptotic\nestimate for the maximal splitting distance, and show the existence of\ntransverse homoclinic orbits to the whiskered tori with an asymptotic estimate\nfor the transversality of the splitting. Both estimates are exponentially small\nin \ε, with the functions in the exponents being periodic with\nrespect to \ln\ε, and can be explicitly constructed from the\ncontinued fraction of \Ω. In this way, we emphasize the strong dependence\nof our results on the arithmetic properties of \Ω. In particular, for\nquadratic ratios \Ω with a 1-periodic or 2-periodic continued fraction\n(called metallic and metallic-colored ratios respectively), we provide accurate\nupper and lower bounds for the splitting. The estimate for the maximal\nsplitting distance is valid for all sufficiently small values of \ε,\nand the transversality can be established for a majority of values of\n\ε, excluding small intervals around some transition values where\nchanges in the dominance of the harmonics take place, and bifurcations could\noccur.\n