An investigation of chaotic diffusion in a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action

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, $\theta$ 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 $ϵ\mathrm{\ne }0$ and; (ii) $\gamma$ denoting the power of the action in the equation defining the angle. For $ϵ\mathrm{\ne }0$ 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.