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

TL;DR

This paper studies chaotic diffusion in a family of Hamiltonian mappings with diverging angles as action vanishes, using diffusion equations and numerical simulations to analyze the transition from integrable to chaotic behavior.

Contribution

It introduces an analytical approach to describe chaotic diffusion in Hamiltonian mappings with diverging angles, validated by numerical simulations.

Findings

Analytical solutions match numerical results well.

Chaotic diffusion depends on control parameters $psilon$ and $mma$.

Transition from integrable to non-integrable dynamics observed.

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, $\theta$ and controlled by two control parameters: (i) $\epsilon$, controlling the nonlinearity of the system, particularly a transition from integrable for $\epsilon=0$ to non-integrable for $\epsilon\ne0$ and; (ii) $\gamma$ denoting the power of the action in the equation defining the angle. For $\epsilon\ne0$ 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.