Comparative study for the calculation of the Lyapunov spectrum from nonlinear experimental signals

A uniform formalism is introduced for the description and comparison of the algorithms of Sane and Sawada [M. Sane and Y. Sawada, Phys. Rev. Lett. 55, 1082 (1985)] and Eckmann et al. [J.-P, Eckmann, S. O. Kamphorst, D. Ruelle, and S. Cilibert, Phys. Rev. A 34, 4971 (1986)], for the calculation of the Lyapunov spectrum from experimental data. It is shown that both algorithms coincide for the calculation of the maximum Lyapunov exponent and differ for the other exponents. A numerical application is carried out which confirms the above result. A detailed investigation of the dependence of the Sane and Sawada and the Eckmann et al. algorithms on the parameters of the algorithms, the signal and the reconstruction of the attractor, for the calculation of the whole Lyapunov spectrum is presented. Calculations are performed for three kinds of signals: (a) the noise-free dynamical variable x(t) of the Lorenz system, (b) the stiff and long duration time evolution of the total current of the electrochemical oscillator Fe-2M H2SO4 in the presence of external Ohmic resistance R, and (c) the smooth Variation and short duration signal of the same experimental system for a different set of parameters. A comparison between the results of the two algorithms is attempted as well as an investigation of the trends of the Lyapunov spectrum by varying the algorithm, signal, and reconstruction parameters.