Diffusion in two-dimensional mappings

Abstract

Two-dimensional mappings are used to model both Hamiltonian and dissipative systems in two degrees of freedom. Numerical iterations of these mappings are then easily performed to reveal the phase space structure, transient and invariant phase space density distributions, and the rate of transport of phase points in “time” (mapping periods). Mappings are chosen for their generic quality, their convenience in analysis, or their close correspondence to physical systems. For Hamiltonian systems (section 2), the self-similar structure of the phase space at all scales leads to long-time correlations of trajectories that decay more slowly than exponentially. The result of these correlations is to modify the quasilinear diffusion rate, which can be calculated to an arbitrary degree of accuracy for mappings which are on a torus (periodic in both action and angle). For generic mappings, in which only the angle is periodic, approximate local diffusion coefficients (averaged over phase) can be obtained and used in a Fokker-Planck equation to determine phase space transport in action. For mappings on a torus, some values of parameters give rise to accelerator modes, which lead to streaming particles for which the diffusion rate may be infinite. For generic mappings which are periodic only in angle, these accelerator modes enhance diffusion but do not lead to singularities. The diffusion coefficient is also obtained for systems in the adiabatic limit in which the small parameter is the ratio of the unperturbed frequencies. For dissipative systems (section 3), transport and phase space distributions are considered both for parameter ranges in which regular attractors exist and parameter ranges having a chaotic (strange) attractor. Particular attention is given to dissipative mappings in which the dissipation is a perturbation from an area-preserving (Hamiltonian) mapping. For small dissipation, the rate of decay of the phase space density, by absorption into regular attractors (sinks) is shown to increase with the dissipation parameter δ in an easily calculable manner. With continued increase in δ, more subtle phenomena appear which lead to a maximum rate of density decay at some δ= δ s and then to a vanishing phase averaged decay rate at some δ= δ cr , beyond which a strange attractor exists. A method of calculating the invariant distribution on the attractor to arbitrary accuracy is described.