Optimization of Poincaré sections for discriminating between stochastic and deterministic behavior of dynamical systems

This paper studies the problem of finding optimal parameters for a Poincaré section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincaré & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincaré section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincaré section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincaré section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincaré section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincaré section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincaré sections for use with the P&H method.