Exponentially Small Splitting of Separatrices Associated to 3D Whiskered Tori with Cubic Frequencies

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 Poincare–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.