ENRM: An alternative tool for studying dynamical systems

In recent years, the recurrence plot technique has been increasingly used in nonlinear complex time series, to analyse the periodicity, chaos and non-stationarity. As far, some important methods have arisen based on the recurrence plot technique, among which the recurrence quantification analysis (RQA) is the most famous one, defined based on the recurrence matrix. As a tool of the RQA, the entropy based on recurrence microstates (ENRM), has been proposed in Corsoet al. (2018), to measure the chaotic and mixing behaviours of the underlying time series. In this paper, we first develop some techniques for more efficiently computing the ENRM. Another important contribution of this paper is to extend the concept of the ENRM to study multi-dimensional dynamical systems, where we regard each particle trajectory as a time series. This allows us to compare the ENRM to the finite time Lyapunov exponent (FTLE), which is another commonly used quantity for measuring chaotic behaviours of multi-dimensional dynamical systems. Examples will show different behaviours of particle trajectories with different magnitudes of ENRM values, and also the difference between the ENRM and the FTLE. Especially, by investigating a real data set, we show that the ENRM is a nice tool for measuring the complex behaviours of dynamical systems in the occasion where only sparse particle trajectories are provided.