Uncertainty exponent
Encyclopedia
In mathematics, the uncertainty exponent is a method of measuring the fractal dimension
of a basin boundary. In a chaotic scattering system, the
invariant set of the system is usually not directly accessible because it is non-attracting and typically of measure
zero. Therefore, the only way to infer the presence of members
and to measure the properties of the invariant set is through the basins of attraction. Note that in a scattering system, basins of attraction are not limit cycles therefore do not constitute members of the invariant set.
Suppose we start with a random trajectory and perturb it by a small amount,
, in a random direction. If the new trajectory ends up
in a different basin from the old one, then it is called epsilon uncertain.
If we take a large number of such trajectories,
then the fraction of them that are epsilon uncertain is the uncertainty fraction,
, and we expect it to scale exponentially
with
:
Thus the uncertainty exponent,
, is defined as follows:
The uncertainty exponent can be shown to approximate the box-counting dimension
as follows:
where N is the embedding dimension. Please refer to the article on chaotic mixing
for an example of numerical computation of the uncertainty dimension
compared with that of a box-counting dimension.
Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
of a basin boundary. In a chaotic scattering system, the
invariant set of the system is usually not directly accessible because it is non-attracting and typically of measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
zero. Therefore, the only way to infer the presence of members
and to measure the properties of the invariant set is through the basins of attraction. Note that in a scattering system, basins of attraction are not limit cycles therefore do not constitute members of the invariant set.
Suppose we start with a random trajectory and perturb it by a small amount,
, in a random direction. If the new trajectory ends upin a different basin from the old one, then it is called epsilon uncertain.
If we take a large number of such trajectories,
then the fraction of them that are epsilon uncertain is the uncertainty fraction,
, and we expect it to scale exponentiallywith
:Thus the uncertainty exponent,
, is defined as follows:The uncertainty exponent can be shown to approximate the box-counting dimension
as follows:
where N is the embedding dimension. Please refer to the article on chaotic mixing
Chaotic mixing
In chaos theory and fluid dynamics, chaotic mixing is a processby which flow tracers develop into complex fractals under the actionof a time-varying fluid flow.The flow is characterized by an exponential growth of fluid filaments....
for an example of numerical computation of the uncertainty dimension
compared with that of a box-counting dimension.