An Alternative Algorithm to Box Counting  

Shinji Koga
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.