4.2 The Complete Ternary Horseshoe
101
The last topic we consider in this chapter (Sect. 4.5), is the dynamics of the
HOCl molecule above the energy for dissociation of the chlorine atom Cl from
the HO complex. We use an experimentally obtained potential energy to analyze
the scattering process (Weiss et al. 2000). Under certain fairly realistic constraints
on the angular momentum of the molecule and on its vibrational dynamics, the
HO–Cl scattering system can be reduced to a 2 DoF system. Above dissociation,
the molecular dynamics is largely chaotic (Barr et al. 2009) and scattering of Cl
from HO has fractal structure (Lin et al. 2015). Finally in Sect. 4.6, we make some
concluding remarks.
4.2 The Complete Ternary Horseshoe
The complete ternary horseshoe provides one of the simplest examples of a fractal
scattering process that involves two dominant unstable fixed points and a fully selfsimilar scattering process. A scattering system governed by a complete ternary
horseshoe can be built from a pair of Gaussian potential energy peaks, each with
one degree of freedom (DoF), to which a periodic “kick” is applied. In the sections
below, we first describe this unperturbed one-dimensional system, and then describe
the effect of a time-periodic delta function kick applied to the system.
4.2.1 Double Gaussian Potential Energy Peaks
The Hamiltonian for a particle that scatters from double Gaussian potential energy
peaks, can be written
H =
p 2
2
+ x
2 e
−x 2 = E,
(4.1)
where the potential energy is constructed so that it goes to zero at x = 0. Hamilton’s
equations, for this system, are given by
˙
p =
dp
dt
= −
∂H
∂x
= 2(x
3
− x)e
−x 2 , ˙
x =
dx
dt
= −
∂H
∂p
= p.
(4.2)
The fixed points of Hamilton’s equations satisfy the conditions ( ˙
p = 0, ˙
x = 0).
There are three fixed points. One fixed point is elliptic at (p = 0, x = 0) and has
energy E = 0. The two remaining fixed points are hyperbolic at (p = 0, x = ±1)
and have energy E = e −1 = 0.3679. A plot of the phase space trajectories for
this system is given in Fig. 4.1. It is clear that these three fixed points determine the
overall structure of the phase space.
101
The last topic we consider in this chapter (Sect. 4.5), is the dynamics of the
HOCl molecule above the energy for dissociation of the chlorine atom Cl from
the HO complex. We use an experimentally obtained potential energy to analyze
the scattering process (Weiss et al. 2000). Under certain fairly realistic constraints
on the angular momentum of the molecule and on its vibrational dynamics, the
HO–Cl scattering system can be reduced to a 2 DoF system. Above dissociation,
the molecular dynamics is largely chaotic (Barr et al. 2009) and scattering of Cl
from HO has fractal structure (Lin et al. 2015). Finally in Sect. 4.6, we make some
concluding remarks.
4.2 The Complete Ternary Horseshoe
The complete ternary horseshoe provides one of the simplest examples of a fractal
scattering process that involves two dominant unstable fixed points and a fully selfsimilar scattering process. A scattering system governed by a complete ternary
horseshoe can be built from a pair of Gaussian potential energy peaks, each with
one degree of freedom (DoF), to which a periodic “kick” is applied. In the sections
below, we first describe this unperturbed one-dimensional system, and then describe
the effect of a time-periodic delta function kick applied to the system.
4.2.1 Double Gaussian Potential Energy Peaks
The Hamiltonian for a particle that scatters from double Gaussian potential energy
peaks, can be written
H =
p 2
2
+ x
2 e
−x 2 = E,
(4.1)
where the potential energy is constructed so that it goes to zero at x = 0. Hamilton’s
equations, for this system, are given by
˙
p =
dp
dt
= −
∂H
∂x
= 2(x
3
− x)e
−x 2 , ˙
x =
dx
dt
= −
∂H
∂p
= p.
(4.2)
The fixed points of Hamilton’s equations satisfy the conditions ( ˙
p = 0, ˙
x = 0).
There are three fixed points. One fixed point is elliptic at (p = 0, x = 0) and has
energy E = 0. The two remaining fixed points are hyperbolic at (p = 0, x = ±1)
and have energy E = e −1 = 0.3679. A plot of the phase space trajectories for
this system is given in Fig. 4.1. It is clear that these three fixed points determine the
overall structure of the phase space.