Fractal Dimensions as Chaos Indicators in the Standardmap
Florian Freistetter
Institut of Astronomy - University of Vienna, Austria
We are using a fractal dimension (the Correlationdimension) to distinguish between chaotic and regular
orbits in the standardmap. This is done by calculating N points for a given initial condition and
assigning a dimension Dn to this orbit. Then, a new point is calculated, resulting in a
dimension Dn+1. Doing this for lots of values, gives the ``time development'' of the
Correlationdimension. And because the different dynamical types develop in a different way, the
dimension development reflects the different features. A property which makes this method
different from most of the others is the fact, that it is possible to detect a ``sticky'' orbit
before it leaves the vincinity of the regular parts in the phase space (at least in most of the
cases). This makes this method very interessting in systems were (a) it is difficult to calculate
lots of points, so that one can distinguish between regular and sticky orbits, or (b) where it is
difficult to represent the motion, i.e. with a Poincaré Surface of Section, like in
high-dimensional systems.