Estimating the Shannon Entropy: Recurrence Plots versus Symbolic Dynamics

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition—still based on the recurrence plots—and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.

Figure 1

Comparison between the largest Lyapunov exponent and the Shannon entropy for the logistic function. One thousand data points are used for computing $S_{\mathrm{RP}}$ (b) and 3000 for $S_{\mathrm{SD}}$ (c).

Figure 2

Shannon entropy estimations depend on different parameters, according to the approach. ($\mu =3.94$)

Figure 3

Comparison between the largest Lyapunov exponent and the Shannon entropy for the logistic function contaminated by a multiplicative noise (0.5%).

Figure 4

Bifurcation diagram (a) of the Rössler system computed over the interval where the dynamics is unimodal. Comparison between the Shannon entropy estimated in a Poincaré section (b) from the recurrence plots and (c) from symbolic dynamics.

Figure 5

Shannon entropy computed using recurrence plots from the different variables of the Rössler system. A 3D phase space is reconstructed using delay coordinates with a time delay $\tau =1.5\mathrm{s}$. Fifteen hundred data points sampled at $\tau$ are used.