Pseudo-boundaries in discontinuous 2-dimensional maps

TL;DR

This paper demonstrates the existence of pseudo-boundaries in discontinuous 2D maps, showing they arise from island chains that influence phase space dynamics, similar to smooth maps.

Contribution

It introduces the concept of pseudo-boundaries in discontinuous maps and explains their origin from island chains affecting phase space separation.

Findings

Pseudo-boundaries are present in discontinuous maps.

Trajectories tend to propagate along these pseudo-boundaries.

The behavior is demonstrated using a generalized Fermi map.

Abstract

It is known that Kolmogorov-Arnold-Moser boundaries appear in sufficiently smooth 2-dimensional area-preserving maps. When such boundaries are destroyed, they become pseudo-boundaries. We show that pseudo-boundaries can also be found in discontinuous maps. The origin of these pseudo-boundaries are groups of chains of islands which separate parts of the phase space and need to be crossed in order to move between the different sub-spaces. Trajectories, however, do not easily cross these chains, but tend to propagate along them. This type of behavior is demonstrated using a ``generalized'' Fermi map.