Using Correlation Integrals to Characterize Three-dimensional Stellar Orbits

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 introduced previously in 1983 by Grassberger & Procaccia and in 1984 by Carnevali & Santangelo 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 (one-, two-, and three-dimensional; 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, three-dimensional 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.