Statistical properties of the maximum Lyapunov exponent calculated via the divergence rate method

The embedding of a time series provides a basic tool to analyze dynamical properties of the underlying chaotic system. To this purpose, the choice of the embedding dimension and lag is crucial. Although several methods have been devised to tackle the issue of the optimal setting of these parameters, a conclusive criterion to make the most appropriate choice is still lacking. An accepted procedure to rank different embedding methods relies on the evaluation of the maximum Lyapunov exponent (MLE) out of embedded time series that are generated by chaotic systems with explicit analytic representation. The MLE is evaluated as the local divergence rate of nearby trajectories. Given a system, embedding methods are ranked according to how close such MLE values are to the true MLE. This is provided by the so-called standard method in a way that exploits the mathematical description of the system and does not require embedding. In this paper we study the dependence of the finite-time MLE evaluated via the divergence rate method on the embedding dimension and lag in the case of time series generated by four systems that are widely used as references in the scientific literature. We develop a completely automatic algorithm that provides the divergence rate and its statistical uncertainty. We show that the uncertainty can provide useful information about the optimal choice of the embedding parameters. In addition, our approach allows us to find which systems provide suitable benchmarks for the comparison and ranking of different embedding methods.