E. Lega et al., On the computation of Lyapunov exponents for discrete time series. Applications to two-dimensional symplectic and dissipative mappings, INT J B CH, 10(12), 2000, pp. 2791-2805
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 B
enettin and Galgani [1979]. The aim of the present paper is to provide an e
xplanation 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 foc
us our attention on conservative systems, which are not widely investigated
in the literature. We show that using standard techniques the results obta
ined for discrete series related to area-preserving mappings are often unre
liable, while dissipative systems are easier to analyze. In order to overco
me the problem arising with conservative systems, we develop an alternative
method, which takes advantage of the existing techniques. In particular, a
ll 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 a
greement with the theoretical expectation for weak chaotic motions, and som
etimes 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.