Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies

TL;DR

This paper derives exponentially small asymptotic estimates for the splitting of invariant manifolds of whiskered tori with quadratic and cubic irrational frequencies in nearly-integrable Hamiltonian systems, highlighting the influence of arithmetic properties.

Contribution

It provides new asymptotic estimates for manifold splitting in Hamiltonian systems with quadratic and cubic irrational frequencies, using continued fractions and specific algebraic properties.

Findings

Asymptotic estimates depend on frequency arithmetic properties

24 quadratic cases satisfy a key arithmetic property

Results for cubic golden number demonstrate similar splitting behavior

Abstract

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems. We consider 2-dimensional tori with a frequency vector $\omega=(1,\Omega)$ where $\Omega$ is a quadratic irrational number, or 3-dimensional tori with a frequency vector $\omega=(1,\Omega,\Omega^2)$ where $\Omega$ is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, showing that such estimates depend strongly on the arithmetic properties of the frequencies. Inthe quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which $\Omega$ is the so-called cubic golden number (the real root of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.