Quantile graphs for the characterization of chaotic dynamics in time series

Recently, a map from time series to networks with an approximate inverse operation has been proposed [1], allowing the use network statistics to characterize time series and time series statistics to characterize networks. In this approach, time series quantiles are mapped into nodes of a graph [1], [2]. Here we show these quantile graphs (QGs) are able to characterize features such as long range correlations or deterministic chaos present in the underlying dynamics of the original signal, making them a powerful tool for the analysis of nonlinear systems. As an illustration we applied the QG method to the Logistic and the Quadratic maps, for varying values of their control parameters. We show that in both cases the main features of resulting bifurcation cascades, with their progressive transition from periodic behavior to chaos, are well captured by the topology of QGs.