Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type

Abstract

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vector $\omega /\sqrt{\epsilon }$, with $\omega =(1,\Omega )$ where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré-Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.

12 pages, 1 figure. arXiv admin note: text overlap with arXiv:1306.0728

References in corpus (2)

• Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation
• Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies

Cited by in corpus (4)

• Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio
• Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
• Splitting of the separatrices after a Hamiltonian-Hopf bifurcation under periodic forcing
• Exponentially small splitting of separatrices near a period-doubling bifurcation in area-preserving maps