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Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods
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10.1063/1.3523304
/content/aip/journal/chaos/20/4/10.1063/1.3523304
http://aip.metastore.ingenta.com/content/aip/journal/chaos/20/4/10.1063/1.3523304

Figures

Image of FIG. 1.
FIG. 1.

Recurrence plots for a (a) periodic and a (b) chaotic trajectory of the Rössler system (9) (see Sec. III B for details).

Image of FIG. 2.
FIG. 2.

Illustration of the shortest path length for two example trajectories of the Rössler system (9) in (a) periodic and (b) chaotic regimes. The square is a schematic projection of the recurrence neighborhood to the plane. In these two particular examples, and for (maximum norm).

Image of FIG. 3.
FIG. 3.

Maximum Lyapunov exponent in the parameter plane of the Rössler system (9). Regions with indicate periodic dynamics, those with large correspond to a strongly chaotic behavior. Asterisks indicate the parameter combinations used as examples in Secs. II and III.

Image of FIG. 4.
FIG. 4.

(a) Probability distributions of mutual distances (maximum norm) between states on one realization of periodic (solid) and chaotic (dashed) trajectories, respectively (see text). (b) Dependence of the recurrence rate RR on the recurrence threshold for a periodic and a chaotic regime. The error bars correspond to the standard deviation obtained from 100 realizations with different initial conditions.

Image of FIG. 5.
FIG. 5.

Dependence of the RQA measures (a) DET and (b) on for periodic (solid) and chaotic (dashed) trajectories. The error bars indicate the standard deviation obtained from 100 realizations of the Rössler system (9) with , , , and different initial conditions.

Image of FIG. 6.
FIG. 6.

Dependence of the RQA measures (a) DET, (b) , and the network measures (c) and (d) on the recurrence threshold for periodic (solid) and chaotic (dashed) trajectories (see text). The error bars indicate the standard deviation obtained from 100 realizations of the Rössler system (9) with , , , and different initial conditions.

Image of FIG. 7.
FIG. 7.

Same as in Fig. 6 for the dependence of these measures on the recurrence rate RR.

Image of FIG. 8.
FIG. 8.

RQA measures (a) DET and (b) and network measures (c) and (d) in the parameter plane of the Rössler system (9).

Image of FIG. 9.
FIG. 9.

CDF differences between the maximum Lyapunov exponent and the RQA measures (a) DET and (b) and network measures (c) and (d) in the parameter plane of the Rössler system (9).

Image of FIG. 10.
FIG. 10.

Probability distribution function of the maximum Lyapunov exponent obtained from all 1 000 000 parameter combinations in the considered plane of the Rössler system (9).

Image of FIG. 11.
FIG. 11.

PDFs of RQA and RN measures for parameter combinations yielding maximum Lyapunov exponents and , respectively, with .

Image of FIG. 12.
FIG. 12.

Measures for the discriminatory skills of the different recurrence-based measures DET (●), (◻), , and obtained from a comparison with the results derived using the maximum Lyapunov exponent in dependence on the choice of : (a) -test statistics, (b) -test statistics, (c) overlap integral [Eq. (12)], and (d) relative frequency of false detections using the same quantiles of and the respective measures.

Image of FIG. 13.
FIG. 13.

(a) ROC curves for and (b) area under the ROC curve (AUC) in dependence on for all four measures. For , AUC takes the values 0.9279 (DET), 0.9090 , 0.9487 , and 0.9442 , respectively.

Image of FIG. 14.
FIG. 14.

Discrimination errors (black dots) for the quantile-based groupings , , , and for (see Sec. ???) for (a) DET , (b) (0.1106), (c) (0.0954), and (d) (0.0899).

Tables

Generic image for table
Table I.

Maximum Lyapunov exponents (, ), mean and maximum separation of points in phase space and resulting recurrence threshold (maximum norm) for , and RQA and network measures for two parameter combinations (see text), representing periodic and chaotic regimes of the Rössler system. The error bars correspond to the standard deviation obtained from 100 realizations with different initial conditions. Note that the large variance of the metric quantities and for the chaotic trajectory is a common result when working with short time series and different initial conditions (Ref. 53).

Generic image for table
Table II.

Overall performance indicators obtained from a point-wise comparison of the values of the maximum Lyapunov exponent and the different RQA and network measures: Spearman’s and the standard deviation of the CDF differences . For simplicity, the arguments of the different characteristics have been omitted.

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/content/aip/journal/chaos/20/4/10.1063/1.3523304
2010-12-08
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods
http://aip.metastore.ingenta.com/content/aip/journal/chaos/20/4/10.1063/1.3523304
10.1063/1.3523304
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