Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies

Abstract

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector $\omega /\sqrt{\epsilon }$, with $\omega =(1,\Omega ,\tilde{\Omega })$ where $\Omega$ is a cubic irrational number whose two conjugates are complex, and the components of $\omega$ generate the field $\mathbb{Q}(\Omega )$. A paradigmatic case is the cubic golden vector, given by the (real) number $\Omega$ satisfying ${\Omega }^3=1-\Omega$, and $\tilde{\Omega }={\Omega }^2$. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors $⟨k,\omega ⟩$, $k\in {\mathbb{Z}}^3$. Applying the Poincaré-Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on $\epsilon$) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in $\epsilon$, and valid for all sufficiently small values of~$\epsilon$. This estimate behaves like $\exp \{-h_1(\epsilon )/{\epsilon }^{1/6}\}$ and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function $h_1(\epsilon )$ in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector $\omega$, and proving that it is a quasiperiodic function (and not periodic) with respect to $\ln \epsilon$. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.

arXiv admin note: text overlap with arXiv:1507.07397