Chaotic scattering problems with polygons and polyhedra: exit basins and uncertainty fractal dimension

Christian, JM

Abstract

The Gaspard-Rice (GR) problem provides a paradigm for studying ballistic scattering in a
plane, and it exhibits the phenomenon of sensitive dependence on initial conditions. The
classic incarnation comprises a point-particle projectile reflecting specularly from three hardedged discs located at the vertices of an equilateral triangle. Between reflections, the
projectile travels at constant velocity. In this presentation, a regular-polygon generalization is
proposed and its impact on properties such as exit basins, their boundaries, and time-delay
functions are explored through computations. Consideration is also given to chaotic
scattering from regular polyhedra, placing particular emphasis on the tetrahedron and
hexahedron with hard spheres centred on their vertices. The ray-tracing algorithm
implemented for GR-type systems can be readily adapted to accommodate motion in three spatial dimensions. Our analysis focuses on the calculation of exit basins, and on probing the
uncertainty fractal dimension of their boundaries as a function of scatterer configuration.
Previous investigations of polyhedra have been primarily concerned with exploring the
topology of basin boundaries through the similarity and box-counting dimensions. Here, we
measure the fraction of phase space associated with uncertain outcomes, allowing us to
establish the susceptibility of these purely deterministic scattering experiments to small
fluctuations in their initial conditions.

Citation

Christian, J. Chaotic scattering problems with polygons and polyhedra: exit basins and uncertainty fractal dimension. Poster presented at 4th IMA Conference on Nonlinearity and Coherent Structures, University of Loughborough (online)

• Presentation Conference Type: Poster
• Conference Name: 4th IMA Conference on Nonlinearity and Coherent Structures
• Conference Location: University of Loughborough (online)
• End Date: Jul 9, 2021
• Publication Date: Jul 9, 2021
• Deposit Date: Jul 12, 2021
• Publicly Available Date: Jul 12, 2021
• Publisher URL: https://cdn.ima.org.uk/wp/wp-content/uploads/2020/10/Abstract-Book-4th-IMA-Conference-on-Nonlinearity-and-Coherent-Structures-2-2.pdf
• Related Public URLs: https://ima.org.uk/15373/4th-ima-conference-on-nonlinearity-and-coherent-structures/
• Additional Information: Event Type : Conference