Christophe LETELLIER
18/05/2009

Recurrence plots were first introduced to quantify the recurrence properties of
chaotic dynamics [1]. A few years later, the recurrence quantification analysis was
introduced to transform graphical representations into statistical analysis [2].
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 has been solved by replacing the probabilities ’s used in the entropy formula

The key point is to replace with the number of diagonal segments of *non-recurrent* points (made of white dots) divided by the number of *recurrent* points [3]. Indeed, a white dot representing a non-recurrent point is nothing more than a signature of complexity within the data. With this definition, increases as the bifurcation parameter increases (as shown for the Logistic map in Fig. 1b). There is a one-to-one correspondence between the new definition of and the positive largest Lyapunov exponent.

**Fig. 1 : Comparison between the Largest Lyapunov Exponent and the Shannon Entropy.**

The algorithm provided here computed the Shannon Entropy from Recurrence Plots for the Logistic map versus parameter . It produced Fig. 1. It is also possible to add noise.

A second Fortran code is provided. It computes the Shannon entropy using a recurrence plot from a data file. you have to specify the number of data point (Npoint). Your data file is expected to have a single column. The code returns

- the recurrence rate ;
- the "determinism" rate ;
- the Shannon entropy.

Be aware that the so-called "determinism" rate does not provide in fact a determinism rate for the simple reason that, for instance, it is equal to 0.82 when estimated from a time series produced by the Logistic map with , a signal which is 100% deterministic ! So, please, interpret with great care this rate. Note also that, for flows, it is definitely better to compute recurrence plots from a "discrete" time series recorded in the Poincaré section of the attractor than from a "continuous" time series. This is discussed in Ref. [3].

[1] **J.-P. Eckmann, S. Oliffson Kamphorst & D. Ruelle**, Recurrence Plots of Dynamical Systems, *Europhysics Letters*, **4**, 973-977, 1987.

[2] **L. L. Trulla, A. Giuliani, J. P. Zbilut & C. L. Webber Jr.**, Recurrence quantification analysis of the logistic equation with transients, *Physics Letters A*, **223** (4), 255-260, 1996.

[3] This latter detail was missing in the definition provided in **C. Letellier**, Estimating the Shannon entropy : recurrence plots versus symbolic dynamics, *Physical Review Letters*, **96**, 254102, 2006.