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

Abstract

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector ! = (1; ), where is a quadratic irrational number, or a 3-dimensional torus with a frequency vector ! = (1; ; 2 ), where is a cubic irrational number. Applying the Poincar e{Melnikov method, we nd exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fullled 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 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 dierences between the results obtained for both the quadratic and cubic cases.