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 $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega,\widetilde\Omega)$ 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 $\widetilde\Omega=\Omega^2$. 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 $\langle k,\omega\rangle$, $k\in{\mathbb Z}^3$. Applying the Poincar\'e-Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on $\varepsilon$) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in $\varepsilon$, and valid for all sufficiently small values of~$\varepsilon$. This estimate behaves like $\exp\{-h_1(\varepsilon)/\varepsilon^{1/6}\}$ and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function $h_1(\varepsilon)$ 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 $\ln\varepsilon$. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.