Exponentially Small Splitting of Separatrices and Transversality Associated to Whiskered Tori with Quadratic Frequency Ratio
2016 (English)In: SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, E-ISSN 1536-0040, Vol. 15, no 2, 981-1024 p.Article in journal (Refereed) Published
The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector w/root epsilon with w = (1, Omega), where the frequency ratio Omega is a quadratic irrational number. Applying the Poincare-Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov potential. This allows us to provide an asymptotic estimate for the maximal splitting distance and show the existence of transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for the transversality of the splitting. Both estimates are exponentially small in epsilon, with the functions in the exponents being periodic with respect to Ins, and can be explicitly constructed from the continued fraction of Omega. In this way, we emphasize the strong dependence of our results on the arithmetic properties of Omega. In particular, for quadratic ratios Omega with a 1-periodic or 2-periodic continued fraction (called metallic and metallic-colored ratios, respectively), we provide accurate upper and lower bounds for the splitting. The estimate for the maximal splitting distance is valid for all sufficiently small values of epsilon, and the transversality can be established for a majority of values of epsilon, excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur.
Place, publisher, year, edition, pages
2016. Vol. 15, no 2, 981-1024 p.
splitting of separatrices, transverse homoclinic orbits, Melnikov integrals, quadratic frequency ratio
IdentifiersURN: urn:nbn:se:uu:diva-308065DOI: 10.1137/15M1032776ISI: 000385262100011OAI: oai:DiVA.org:uu-308065DiVA: diva2:1050991
FunderKnut and Alice Wallenberg Foundation, 2013-0315