On a Very Steep Version of the Standard Map

TL;DR

This paper studies the long-term behavior of a discontinuous version of the standard Chirikov map, showing bounded trajectories for some parameters and estimating escape rates for others, supported by numerical evidence.

Contribution

It introduces a discontinuous variant of the standard map and analyzes its long-term dynamics, including boundedness and escape rate estimates, with numerical support.

Findings

Trajectories remain bounded for certain parameters.

Escape rates can be estimated for other parameters.

Numerical evidence supports the conjecture on escape rates.

Abstract

We consider the long time behavior of the trajectories of the discontinuous analog of the standard Chirikov map. We prove that for some values of parameters all the trajectories remains bounded for all time. For other set of parameters we provide an estimate for the escape rate for the trajectories and present a numerically supported conjecture for the actual escape rate.