Elucidating the escape dynamics of the four hill potential

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 threedimensional 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.

Escape of a particle from two-dimensional potential well

“…For the case of two DOF, former work investigated the escape of two coupled particles from one dimensional potential well [16,17], and suggested an analytically approximated prediction of the escape. When the escape of a particle from a two-DOF potential well is considered, one encounters many detailed numerical investigations [18][19][20][21]. In this work, we extend the discussion and suggest an analytical framework for this type of problems.…”

Final state sensitivity in noisy chaotic scattering

“…For comparison purposes, we have also used the four-hill system [36,37], whose potential consists of four hills located at (x, y) = (±1, ±1) and its Hamiltonian is given by:…”

Escaping from a degenerate version of the four hill potential

“…The system has also been used to illustrate some properties of open systems in section 6.3.2.1 in [24]. In a recent paper [45] (hereafter Paper I) we presented the dynamics of escape of the four hill potential on both the configuration and the phase spaces. In the current work we will put emphasis on the basins of escape and present the complicated basin structure, by using modern color-coded plots, over various 2 dimensional planes of initial conditions.…”