  In an effort to more fully understand the variety of stellar distribution
functions that can be used to construct models of realistic galaxies, the
correlation integral method for orbit characterization that previously has been
introduced by Grassberger & Procaccia (1983) and Carnevali & Santangelo (1984)
is examined in considerable detail. The broad utility of the method is
validated and demonstrated by applying it to orbits in a number of different,
previously studied test cases (1D, 2D, and 3D; nonrotating and rotating). At
the same time, the correlation integral method is compared and contrasted with
other more traditional characterization tools such as Lyapunov exponents and
surfaces of section. The method is then extended to orbits in a previously
unexamined rotating, 3D bar potential. The correlation integral method is found
to be a simple and reliable way to quantitatively categorize orbits in
virtually any potential. It is recommended that it be broadly adopted as a tool
for characterizing the properties of orbits and, by extension, stellar
distribution functions, in all Hamiltonian dynamical systems.
