100
4 Chaotic Scattering
a signature of the type of chaotic dynamics occurring in the reaction region. Fully
developed chaos can imprint fairly simple fractal patterns and statistical properties.
Mixed phase spaces in the reaction region, a more typical situation, present a
much more complex relation between the dynamics in the reaction region and the
asymptotic dynamics of the scattering system.
A pathway for analyzing the complex scattering patterns of a chaotic system
was introduced by Smale (1967), who showed that a mapping of the phase space
flow, in the neighborhood of unstable fixed points of chaotic systems, evolves in a
horseshoe-like fractal folding pattern. Following the work of Smale, methods for
extracting information about chaotic scattering processes and the underlying fractal
structure that governs the dynamics, has been developed by Jung and others for a
variety of model systems (Jung and Scholz 1988; Jung and Richter 1990; Rückerl
1994; Jung et al. 1999). In subsequent sections, we first describe the behavior of a
perfect scattering system, one whose scattering is governed by a complete ternary
horseshoe. We then show examples of chaotic scattering processes in a variety of
systems that can be found in nature.
A scattering system whose chaotic reaction region is describable by a complete
ternary horseshoe is described in Sect. 4.2. The system involves a 1D potential
kicked periodically in time with a delta-function kick, and it contains two primary
unstable fixed points. It is possible to observe a fractal pattern of singularities in
the scattering cross section and to map out a complete symbolic dynamics for the
scattering process. This analysis will allow us to introduce concepts of stable and
unstable manifolds associated to the fixed points, heteroclinic intersections, and
show how the influence of these manifolds extends far into the asymptotic region.
It also allows us to obtain the fractal structure of the delay time for transmission
and reflection of incident particles, and more generally the fractal pattern associated
with all dynamical functions that involve the scattering process.
Most real systems do not involve a complete horseshoe, because the reaction
region contains a phase space that is a self-similar mixture of regular and chaotic
structures. Then the pattern of singularities in the scattering functions is not as
simple. When there is a mixed phase space, a mixture of chaos, periodic orbits,
and KAM islands can exist in the continuum and trajectories starting on a periodic
orbit or KAM island will stay in the reaction region forever.
The first example of a scattering system with an incomplete horseshoe, is that
of a charged particle scattering from a magnetic dipole, considered in Sect. 4.3.
Jung and Scholz (1987, 1988) found a direct relation between homoclinic orbits
in the reaction region and the structure of the delay time for scattered particles.
A slightly more complex scattering process, considered in Sect. 4.4, involves the
scattering of an electron from a negative chlorine ion Cl − in the presence of a
monochromatic radiation field (Emmanouilidou et al. 2003; Jung and Emmanouilidou 2005; Emmanouilidou and Jung 2006; Lin et al. 2011). This system has a
fractal distribution of delay times that can be described by a symbolic dynamics.
The symbolic dynamics can be described in terms of a branching tree, which, in
turn, allows construction of a transfer matrix. Eigenvectors of the transfer matrix
give us information about the fraction of incident particles that are transmitted or
reflected in the scattering process.
4 Chaotic Scattering
a signature of the type of chaotic dynamics occurring in the reaction region. Fully
developed chaos can imprint fairly simple fractal patterns and statistical properties.
Mixed phase spaces in the reaction region, a more typical situation, present a
much more complex relation between the dynamics in the reaction region and the
asymptotic dynamics of the scattering system.
A pathway for analyzing the complex scattering patterns of a chaotic system
was introduced by Smale (1967), who showed that a mapping of the phase space
flow, in the neighborhood of unstable fixed points of chaotic systems, evolves in a
horseshoe-like fractal folding pattern. Following the work of Smale, methods for
extracting information about chaotic scattering processes and the underlying fractal
structure that governs the dynamics, has been developed by Jung and others for a
variety of model systems (Jung and Scholz 1988; Jung and Richter 1990; Rückerl
1994; Jung et al. 1999). In subsequent sections, we first describe the behavior of a
perfect scattering system, one whose scattering is governed by a complete ternary
horseshoe. We then show examples of chaotic scattering processes in a variety of
systems that can be found in nature.
A scattering system whose chaotic reaction region is describable by a complete
ternary horseshoe is described in Sect. 4.2. The system involves a 1D potential
kicked periodically in time with a delta-function kick, and it contains two primary
unstable fixed points. It is possible to observe a fractal pattern of singularities in
the scattering cross section and to map out a complete symbolic dynamics for the
scattering process. This analysis will allow us to introduce concepts of stable and
unstable manifolds associated to the fixed points, heteroclinic intersections, and
show how the influence of these manifolds extends far into the asymptotic region.
It also allows us to obtain the fractal structure of the delay time for transmission
and reflection of incident particles, and more generally the fractal pattern associated
with all dynamical functions that involve the scattering process.
Most real systems do not involve a complete horseshoe, because the reaction
region contains a phase space that is a self-similar mixture of regular and chaotic
structures. Then the pattern of singularities in the scattering functions is not as
simple. When there is a mixed phase space, a mixture of chaos, periodic orbits,
and KAM islands can exist in the continuum and trajectories starting on a periodic
orbit or KAM island will stay in the reaction region forever.
The first example of a scattering system with an incomplete horseshoe, is that
of a charged particle scattering from a magnetic dipole, considered in Sect. 4.3.
Jung and Scholz (1987, 1988) found a direct relation between homoclinic orbits
in the reaction region and the structure of the delay time for scattered particles.
A slightly more complex scattering process, considered in Sect. 4.4, involves the
scattering of an electron from a negative chlorine ion Cl − in the presence of a
monochromatic radiation field (Emmanouilidou et al. 2003; Jung and Emmanouilidou 2005; Emmanouilidou and Jung 2006; Lin et al. 2011). This system has a
fractal distribution of delay times that can be described by a symbolic dynamics.
The symbolic dynamics can be described in terms of a branching tree, which, in
turn, allows construction of a transfer matrix. Eigenvectors of the transfer matrix
give us information about the fraction of incident particles that are transmitted or
reflected in the scattering process.