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

## TL;DR

This paper investigates the exponentially small splitting of invariant manifolds of 3D whiskered tori with cubic irrational frequencies in nearly-integrable Hamiltonian systems, developing new methods to analyze small divisors and providing precise asymptotic estimates.

## Contribution

It introduces a novel methodology for analyzing small divisors with cubic irrational frequencies and derives explicit asymptotic estimates for the splitting, including the quasiperiodic behavior of the splitting function.

## Key findings

- Splitting distance is exponentially small, like exp ext{-}ig\{-h_1( ext{epsilon})/ ext{epsilon}^{1/6}ig\}
- The function h_1(epsilon) is explicitly constructed from resonance properties and is quasiperiodic in ln(epsilon)
- Developed a new approach for small divisor analysis in systems with cubic irrational 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.