ACTION DIFFUSION FOR SYMPLECTIC MAPS WITH A NOISY LINEAR FREQUENCY

We consider an area preserving map in the neighbourhood of an elliptic fixed point, whose linear frequency is stochastically perturbed. The nonlinearity couples the random motion in the phase with the action wh ich exhibits a diffusive behaviour. If the unperturbed dynamics is alm ost integrable and no macroscopic resonant structures are present in t he phase space, a Fokker-Planck equation for the action diffusion is d erived and its solution shows an excellent agreement with the simulati on of the process. The key points are the description of the unperturb ed motion by using the normal forms and the derivation of a stochastic ally perturbed interpolating Hamiltonian for which the action diffusio n coefficient is analytically computed. The angle averaging is justifi ed by the much faster time scale on which the angle relaxes to a unifo rm distribution.