PARAMETRIC RESURGENCE AND EXPONENTIAL SMALLNESS OF THE SPLITTING OF THE SEPARATRICES OF THE RAPIDLY FORCED PENDULUM

Henri Poincare had already noticed that the stable and unstable manifolds of the perturbed pendulum defined by the Hamiltonian H(q, p, t) = p(2)/2 + (-1 + cos q)(1 - mu sin(t/epsilon)), do not coincide when parameter mu is not equal to zero, and that the same formal divergent series in powers of epsilon may be associated with both of them. Here this divergence is analyzed by means of the recent theory of resurgence and alien calculus which allows to estimate asymptotically the size of the splitting of the manifolds as epsilon tends to zero - at least this is proven for the simplified problem where sin(t/epsilon) is replace with e(it)/(epsilon).