Periodic orbits
Topics:
Symbolic dynamics,
Inverse quantum chaology,
General approach to symbolic dynamics,
Bifurcations
Symbolic dynamics
The Gutzwiller trace formula provides a fundamental relationship
between the quantum mechanical density of states and
a sum over all periodic orbits of the classical system.
Thus it is of vital importance to have a detailed knowledge
of the periodic orbits of the system; usually this is
achieved by finding a
symbolic dynamics, i.e.
an encoding of the dynamical system in terms of a (preferably)
finite alphabet.
In the paper
Symbolic dynamics and periodic orbits
for the cardioid billiard
a symbolic dynamics for the cardioid billiard with just two symbols
was established. This allowed to determine all periodic
orbits with up to 20 reflections, e.g.:
Inverse quantum chaology
Using a Fourier transform of the quantum mechanical density
of states the periodic orbits show up as peaks
at the places of their geometric length
L:
See the paper
Spectral statistics in the
quantized cardioid billiard
for further details.
General approach to symbolic dynamics in billiards
More generally, one can prove that in a wide class of hyperbolic
billiards with singularities a symbolic dynamics with
a finite number of symbols can be found,
see the paper
Generating partitions for
two-dimensional hyperbolic maps for
details.
The result for example applies to the cardioid billiard,
dispersing billiards and the stadium billiard.
For the stadium billiard a symbolic dynamics with
16 symbols is obtained. Fortunately, this can
be reduced to one with 5 symbols.
Periodic orbits and bifurcations in mixed systems
In systems with mixed phase space periodic orbits
can bifurcate when one varies some external parameter.
For the family of limaçon billiards this was studied
in detail in the paper
About ergodicity
in the family of limaçon billiards.
In addition to the question on the density of bifurcations
and stability intervals it was demonstrated that
arbitrarily close to the cardioid one can find stable periodic orbits.
Parameter of creation of AnBB orbits:
Example of an island close to the cardioid at ε=0.93395:
References:
Last modified: 27 October 2004, 16:21:54
Impressum, © Arnd Bäcker
