A comparison between methods to compute Lyapunov exponents

Lyapunov characteristic exponents measure the rate of exponential divergence between neighboring trajectories in the phase space. For a given autonomous dynamical system, the maximum Lyapunov characteristic exponent (hereafter LCE) is computed from the solution of the variational equations of the system. There are many dynamical systems in which the formulation and solution of the variational equations is a cumbersome task. In those cases an alternative procedure, first introduced by Benettin et al., is to replace the variational solution by computing two neighbor trajectories (the test particle and its shadow) and calculating the mutual distance. In this paper, we deal with a comparison between these two different techniques for the calculation of LCE: the variational method and the two-particle method. We point out a problem that can appear when the two-particle method is used, which can lead to a false estimation of a positive LCE. The explanation of this phenomenon can be analyzed in two different situations: (1) for relatively large initial separations the two-particle method is not a good approximation to the solution of the variational equations, and (2) for small initial separations the two-particle method have problems related to the machine precision, even when the separation can be many order of magnitudes larger than the machine precision. We show some examples of false estimates of the LCE that have already appeared in the literature using the two-particle method, and finally we present some suggestions to be taken into account when this method has to be used.