Hamiltonian transport on unstable periodic orbits

Abstract

Transport in Hamiltonian systems is approached on the basis of the unstable periodic orbits (UPOs) embedded in the chaotic region. Correlation functions and diffusion rates are defined for UPO ensembles of two-dimensional periodic maps. Global diffusion in these maps is interpreted in terms of the ensembles U T , containing all the UPOs of period T: as T → ∞, the relative populations of closed orbits and accelerator modes in U T approach a Gaussian distribution, characterized by the diffusion coefficient D. Evidence for the validity of this mechanism is provided for the cat and sawtooth maps. Prime-lattice ensembles of the cat maps appear to be optimal in reproducing the global diffusion. These ensembles are, generically, small subsets of U T for some T. It is shown, however, that the corresponding diffusion rates assume, for T odd, precisely the known value of D in the entire time interval t ≤ T. For T even, all orbits are closed, and D is reproduced only in the interval t ≤ 1 2 T . The diffusion rates associated with ensembles U T of the sawtooth maps appear to approximate well the value of D if T is of the order of the diffusion time. The case of local transport rates, associated with ensembles that are localized in phase space, is illustrated.