Distance correlation detecting Lyapunov instabilities, noise-induced escape times and mixing

The properties of the statistical method of distance correlation between multivariate data are analysed in the context of nonlinear dynamical systems. The distance correlation between the noisy and the noiseless quadratic maps are studied in periodic and chaotic regimes. Results are compared to the classical method of Pearson's correlation. While distance and Pearson's correlations are affected by the Lyapunov exponent from the noiseless orbits, only the distance correlation is able to recognize the correct qualitative behaviour of noise-induced escape times decays and the mixing of chaotic trajectories. In addition, the distance correlation is capable of detecting distinct size of attractors. The main goal of this work is to establish the validity of the distance correlation as an method of correlation between multivariate data in dynamical systems.