# Elucidating the escape dynamics of the four hill potential

## TL;DR

This paper investigates the escape dynamics of the four hill potential through extensive numerical analysis, revealing that all initial conditions lead to escape and examining the fractal nature of escape basins.

## Contribution

It provides a detailed numerical study of escape mechanisms in the four hill potential, highlighting the evolution of fractality and the correlation with escape times.

## Key findings

- All initial conditions lead to escape, with no stable bounded orbits.
- Escape times are longer near fractal basin boundaries.
- Fractality of escape basins varies with energy levels.

## Abstract

The escape mechanism of the four hill potential is explored. A thorough numerical investigation takes place in several types of two-dimensional planes and also in a three-dimensional subspace of the entire four-dimensional phase space in order to distinguish between bounded (ordered and chaotic) and escaping orbits. The determination of the location of the basins of escape toward the different escape channels and their correlations with the corresponding escape time of the orbits is undoubtedly an issue of paramount importance. It was found that in all examined cases all initial conditions correspond to escaping orbits, while there is no numerical indication of stable bounded motion, apart from some isolated unstable periodic orbits. Furthermore, we monitor how the fractality evolves when the total orbital energy varies. The larger escape periods have been measured for orbits with initial conditions in the fractal basin boundaries, while the lowest escape rates belong to orbits with initial conditions inside the basins of escape. We hope that our numerical analysis will be useful for a further understanding of the escape dynamics of orbits in open Hamiltonian systems with two degrees of freedom.