Distinguishing dynamics using recurrence-time statistics

The probability densities of the mean recurrence time, which is the average time needed for a system to recur to a previously visited neighborhood, are investigated in various dynamical regimes and are found to be in agreement with those of the finite-time Lyapunov exponents. The important advantages of the former ones are that they are easy to estimate and that comparable short time series are sufficient. Asymmetric distributions with exponential tails are observed for intermittency and crisis-induced intermittency, while for typical chaos, the distribution has a Gaussian shape. Further, the shape of the distribution distinguishes intermittent strange nonchaotic attractors from those appearing through fractalization and tori collision mechanisms. Furthermore, statistics performed on the peaks in the frequency distribution of recurrence times unveil scaling behavior in agreement with that obtained from the spectral distribution function defined as the number of peaks in the Fourier spectrum greater than a predefined value. The results of the present recurrence statistics are of relevance in classifying different dynamics and providing important insights into the dynamics of a system when only one realization of this system is available. The practical use of this approach for experimental data is shown on experimental electrochemical time series.

Figure 1

RP illustrating the estimators of RTs (black arrows), based on (a) distance between starting points [6] and (c) white vertical lines [5]. For vertically extended recurrence structures, both estimators differ and present upper and lower limits of the Poincaré RT, which would be actually the distance between the midpoints of the recurrence structures (e). (b), (d), (f) Recurrence points of a state $i$ illustrating the differences in the estimators for the RT $T$. (b) The estimator ${\tilde{T}}_G$ as defined by Gao [6] corresponds to the time distance between the first points falling into the neighborhood of state $i$ (here $i-2$ and $i+{\tilde{T}}_G-2$). (d) The estimator ${\tilde{T}}_W$ as defined by white vertical lines [5] corresponds to the time distance between the last and first points falling into the neighborhood of state $i$ (here $i+2$ and $i+{\tilde{T}}_W+2$). (f) The Poincaré RT $T$ for the illustrated case.

Figure 2

Probability density of (a) MRTs and (b) FTLEs, computed from the logistic map Eq. (3), with $\alpha =3.7$ giving rise to a typical chaos and the threshold $\delta =0.1\sigma$, where $\sigma$ is the standard deviation of the data. For each case a Gaussian (dashed red line) is fitted on the distribution.

Figure 3

Variance of MRTs in dependence on $N$ and $\delta$ for a typical chaos in the logistic map Eq. (3), with $\alpha =3.7$. All the axes are in log scale.

Figure 4

Illustration of the decay of the variance of MRTs observed in Fig. 3 for (a) fixed thresholds $\delta =0.03\sigma$, $\delta =0.06\sigma$, and $\delta =0.1\sigma$ as $N$ varies and (b) fixed $N=2000$, $N=4000$, and $N=8000$ as $\delta$ varies. Both axes are in log scale.

Figure 5

Probability density of (a) MRTs and (b) FTLEs, computed using the threshold $\delta =0.1\sigma$ and data from the logistic map Eq. (3), with $\alpha =1+\sqrt{8}-{10}^{-6}$ for which type-I IT is observed.

Figure 6

Probability density of (a) MRTs and (b) FTLEs, computed using the threshold $\delta =0.1\sigma$ and data from the logistic map Eq. (3), with $\alpha =3.7447104$ for which an interior crisis occurs.

Figure 7

Distribution of MRTs (a), (c), (e) and FTLEs (b), (d), (f) through transitions to SNAs in the logistic map Eq. (4). (a), (b) FT route, ${\varepsilon }^{\prime }=1$, $\alpha =2.63$ for torus, and $\alpha =2.66$ for SNA; (c), (d) HH route, ${\varepsilon }^{\prime }=0.3$, $\alpha =3.4874$ for torus, and $\alpha =3.488$ for SNA; (e), (f) IT route, ${\varepsilon }^{\prime }=1$, $\alpha =3.40581$ for torus, and $\alpha =3.4058056$ for SNA.

Figure 8

Distributions of MRTs (a), (c), (e) and FTLEs (b), (d), (f) through transitions to SNAs in the system Eq. (5). (a), (b) FT route; (c), (d) HH route; (e), (f) IT route.

Figure 9

Number of peaks $N\left(S\right)$ in the frequency distribution of RTs which are greater than a predefined threshold $S$ (circles). (a) Two-frequency quasiperiodic motion, and (b) SND. The solid line in (a) corresponds to the scaling relation $N\left(S\right)\sim \text{ln}(1/S)$ and in (b) to the scaling relation $N\left(S\right)\sim S^{-\gamma }$, with $\gamma =0.39\pm 0.0003$. The plots are shown in semilog and log-log axes for (a) and (b), respectively.

Figure 10

Probability density of MRTs computed from experimental electrochemical data (black solid line): (a) quasiperiodic motion; (b) SNA. The dashed red line is a Gaussian fit.

Figure 11

From experimental data: number of peaks $N\left(S\right)$ in the frequency distribution of RTs which are greater than a predefined threshold $S$ (circles). (a) Quasiperiodic motion; (b) SNA. The solid line in (a) corresponds to the scaling relation $N\left(S\right)\sim \text{ln}(1/S)$ and in (b) to the scaling relation $N\left(S\right)\sim S^{-\gamma }$, with $\gamma =0.69\pm 0.007$. The plots are shown in semilog and log-log axes for (a) and (b), respectively.