An Investigation of Chaotic Diffusion in a Family of Hamiltonian Mappings Whose Angles Diverge in the Limit of Vanishingly Action

Abstract

The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, I, and angle, θ and controlled by two control parameters: (i) ϵ, controlling the nonlinearity of the system, particularly a transition from integrable for ϵ= 0 to non-integrable for ϵ≠ 0 and; (ii) γ denoting the power of the action in the equation defining the angle. For ϵ≠ 0 the phase space is mixed and chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures.