Appropriateness of dynamical systems for the comparison of different embedding methods via calculation of the maximum Lyapunov exponent

The embedding of time series provides a valuable, and sometimes indispensable, tool in order to analyze the dynamical properties of a chaotic system. To this purpose, the choice of the embedding dimension and lag is decisive. The scientific literature describes several methods for selecting the most appropriate parameter pairs. Unfortunately, no conclusive criterion to decide which method-and thus which embedding pair-is the best has been so far devised. A widely employed quantity to compare different methods is the maximum Lyapunov exponent (MLE) because, for chaotic systems that have explicit analytic representations, MLE can be numerically evaluated independently of the embedding dimension and lag. Within this framework, we investigated the dependence on the calculated MLE on the embedding dimension and lag in the case of three dynamical systems that are also widespreadly used as reference systems, namely the Lorenz, Rössler and Mackey-Glass attractors. By also taking into account the statistical fluctuations of the calculated MLE, we propose a new method to assess which systems provide suitable test benches for the comparison of different embedding methods via MLE calculation. For example we found that, despite of its popularity in this scientific context, the Rössler attractor is not a reliable workbench to test the validity of an embedding method.