An Alternative Algorithm to Box Counting
Osaka Kyoiku University
By means of Mathematica programming, we construct algorithms enabling us to
evaluate various statistical quantities such as moments, correlation function matrices,
and Lyapunov spectra in n-dimensional chaotic systems with discrete time steps.
These algorithms are fundamentally based on a closed form of an integral representation of
an approximate probability density, which holds as long as a Frobenius-Perron operator is
asymptotically stable; in other words, the system is mixing. Therefore, our method is
basically different from a usual box counting. After introducing our algorithms for
calculating statistical quantities, we compare our method with the box counting by
examining the Hénon map as a concrete example. Throughout our study, our emphasis is on
the extent to which our method enables us to investigate the "hyperfine"
structure of the Hénon attractor.