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Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Universitat Politècnica de Catalunya. Departament de Matemàtiques; Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics
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 $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega),$ where the frequency ratio $\Omega$ is a quadratic irrational number. Applying the Poincaré--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 $\varepsilon$, with the functions in the exponents being periodic with respect to $\ln\varepsilon$, 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 $\varepsilon$, and the transversality can be established for a majority of values of $\varepsilon$, excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur. Read More:
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística
Hamiltonian systems
Sistemes hamiltonians
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics

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