On the computation of Lyapunov exponents for discrete time series. Applications to two dimensional symplectic and dissipative mappings

Many techniques have been developed for the measure of the largest Lyapunov exponent of experimental short data series. The main idea, underlying the most common algorithms, is to mimic the method of computation proposed by Benettin and Galgani [1979]. The aim of the present paper is to provide an explanation for the reliability of some algorithms developed for short time series. To this end, we consider two-dimensional mappings as model problems and we compare the results obtained using the Benettin and Galgani method to those obtained using some algorithms for the computation of the largest Lyapunov exponent when dealing with short data series. In particular we focus our attention on conservative systems, which are not widely investigated in the literature. We show that using standard techniques the results obtained for discrete series related to area-preserving mappings are often unreliable, while dissipative systems are easier to analyze. In order to overcome the problem arising with conservative systems, we develop an alternative method, which takes advantage of the existing techniques. In particular, all algorithms provide a good approximation of the largest Lyapunov exponent in the strong chaotic symplectic case and in the dissipative one. However, the application of standard algorithms provides results which are not in agreement with the theoretical expectation for weak chaotic motions, and sometimes also for regular orbits. On the contrary, the method that we propose in this paper seems to work well for the weak chaotic case. Because of the speed of computation, we suggest to use all algorithms to cross-check the results.