Chapter 4
Chaotic Scattering
Abstract Phase space flow in the neighborhood of unstable fixed points evolves
in a horseshoe-like fractal folding pattern. When incident trajectories scatter from
a localized chaotic object, the outgoing scattered trajectories are described by
functions that have fractal patterns of singularities characteristic of the chaotic
scattering region. Fully developed chaos can imprint fairly simple fractal patterns.
Scattering regions with mixed dynamics have a much more complex asymptotic
dynamics.
An example of a perfect scattering system, is one whose scattering dynamics is
governed by a complete ternary horseshoe. There is a complete fractal pattern of
singularities in the scattering cross section and delay time, and a complete symbolic
dynamics exists for the scattering process.
Some examples of chaotic scattering processes that can be found in nature
include scattering of a charge particle from a magnetic dipole and scattering of an
electron from a negative chlorine ion Cl − . in the presence of a monochromatic
radiation filed. Another example is the dynamics involved in the dissociation of
molecules, such as that of the HOCl molecule above the energy for dissociation
of the chlorine atom Cl from the HO complex. Experimentally obtained potential
energy surfaces can be used to analyze the dissociation (and scattering) process.
Above dissociation, the molecular dynamics is largely chaotic and scattering of Cl
from HO has fractal structure.
Keywords Chaotic scattering · Ternary horseshoe · Fractal scattering patterns ·
Symbolic dynamics · Delay time · Scattering map · Magnetic dipole · Chlorine
ion in radiation field · HOCl dissociation dynamics · Normally hyperbolic
invariant manifold · Galaxies
4.1 Introduction
For scattering systems, the detailed structure of the chaotic dynamics in the reaction
region determines the structure of fractal patterns of singularities imprinted on
classical scattering functions in the asymptotic region. These fractal patterns are
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_4
99
Chaotic Scattering
Abstract Phase space flow in the neighborhood of unstable fixed points evolves
in a horseshoe-like fractal folding pattern. When incident trajectories scatter from
a localized chaotic object, the outgoing scattered trajectories are described by
functions that have fractal patterns of singularities characteristic of the chaotic
scattering region. Fully developed chaos can imprint fairly simple fractal patterns.
Scattering regions with mixed dynamics have a much more complex asymptotic
dynamics.
An example of a perfect scattering system, is one whose scattering dynamics is
governed by a complete ternary horseshoe. There is a complete fractal pattern of
singularities in the scattering cross section and delay time, and a complete symbolic
dynamics exists for the scattering process.
Some examples of chaotic scattering processes that can be found in nature
include scattering of a charge particle from a magnetic dipole and scattering of an
electron from a negative chlorine ion Cl − . in the presence of a monochromatic
radiation filed. Another example is the dynamics involved in the dissociation of
molecules, such as that of the HOCl molecule above the energy for dissociation
of the chlorine atom Cl from the HO complex. Experimentally obtained potential
energy surfaces can be used to analyze the dissociation (and scattering) process.
Above dissociation, the molecular dynamics is largely chaotic and scattering of Cl
from HO has fractal structure.
Keywords Chaotic scattering · Ternary horseshoe · Fractal scattering patterns ·
Symbolic dynamics · Delay time · Scattering map · Magnetic dipole · Chlorine
ion in radiation field · HOCl dissociation dynamics · Normally hyperbolic
invariant manifold · Galaxies
4.1 Introduction
For scattering systems, the detailed structure of the chaotic dynamics in the reaction
region determines the structure of fractal patterns of singularities imprinted on
classical scattering functions in the asymptotic region. These fractal patterns are
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_4
99