^{1}, Reik V. Donner

^{1,2}, Jonathan F. Donges

^{1,3}, Norbert Marwan

^{1}and Jürgen Kurths

^{1,3,4}

### Abstract

The identification of complex periodic windows in the two-dimensional parameter space of certain dynamical systems has recently attracted considerable interest. While for discrete systems, a discrimination between periodic and chaotic windows can be easily made based on the maximum Lyapunov exponent of the system, this remains a challenging task for continuous systems, especially if only short time series are available (e.g., in case of experimental data). In this work, we demonstrate that nonlinear measures based on recurrence plots obtained from such trajectories provide a practicable alternative for numerically detecting shrimps. Traditional diagonal line-based measures of recurrence quantification analysis as well as measures from complex network theory are shown to allow an excellent classification of periodic and chaotic behavior in parameter space. Using the well-studied Rössler system as a benchmark example, we find that the average path length and the clustering coefficient of the resulting recurrence networks are particularly powerful discriminatory statistics for the identification of complex periodic windows.

The investigation of the qualitative behavior in the full parameter space of a complex system is a very important but often challenging task. Detailed knowledge about the different possible types of dynamics helps in understanding under which conditions the particular states of a system lose stability or undergo significant qualitative changes. In particular, in experimental settings, the availability of information about the underlying patterns in phase space allows tuning the critical parameters in such a way that one may obtain the desired working conditions. Mathematically, the corresponding problem is traditionally investigated by means of bifurcation theory, which allows studying the properties of dynamical transitions in some detail.

^{1,2}However, the applicability of available methods for identifying bifurcation scenarios and determining the parameters at which they take place does often depend on the considered system itself. This is especially the case when dealing with larger sets of parameters, i.e., operating in a two- or even higher-dimensional parameter space, in particular for the case of experimental data. In this work, we propose some methods based on recurrence properties in phase space that allow quantifying dynamically relevant properties from available time series, which we harness to disentangle the labyrinthine parameter space with respect to qualitatively and quantitatively different dynamics.

This work has been financially supported by the German Research Foundation (DFG) (Project No. He 2789/8-2), the Max Planck Society, the Federal Ministry for Education and Research (BMBF) via the Potsdam Research Cluster for Georisk Analysis, Environmental Change and Sustainability (PROGRESS), and the Leibniz association (project ECONS). All complex network measures have been calculated using the software package IGRAPH.^{66} We thank K. Kramer for help with the IBM iDataPlex Cluster at the Potsdam Institute for Climate Impact Research, on which all calculations were performed.

I. INTRODUCTION

II. METHODS

A. RQA

B. Quantitative analysis of recurrence networks

III. MODEL: CHAOTIC RÖSSLER SYSTEM

A. Two-dimensional parameter space

B. Prototypical trajectories

C. Dependence on

D. Dependence on the recurrence threshold

IV. RESULTS

A. Behavior of individual measures

B. Correlations between recurrence-based measures and

C. Probability of classification errors

1. and tests

2. Group overlap for fixed probability quantiles

3. Receiver operating characteristics analysis

V. CONCLUSIONS

### Key Topics

- Chaotic dynamics
- 24.0
- Time series analysis
- 21.0
- Attractors
- 11.0
- Bifurcations
- 11.0
- Cluster analysis
- 8.0

## Figures

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

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

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).

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).

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.

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.

(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.

(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.

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.

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.

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.

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.

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

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

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

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

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).

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).

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).

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

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

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.

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.

(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.

(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.

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).

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

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).

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).

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.

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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