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### Abstract

Recently, Wiebe and Virgin [Chaos 22, 013136 (2012)] developed an algorithm which detects chaos by analyzing a time series' power spectrum which is computed using the Discrete Fourier Transform (DFT). Their algorithm, like other time series characterization algorithms, requires that the time series be regularly sampled. Real-world data, however, are often irregularly sampled, thus, making the detection of chaotic behavior difficult or impossible with those methods. In this paper, a characterization algorithm is presented, which effectively detects chaos in irregularly sampled time series. The work presented here is a modification of Wiebe and Virgin's algorithm and uses the Lomb-Scargle Periodogram (LSP) to compute a series' power spectrum instead of the DFT. The DFT is not appropriate for irregularly sampled time series. However, the LSP is capable of computing the frequency content of irregularly sampled data. Furthermore, a new method of analyzing the power spectrum is developed, which can be useful for differentiating between chaotic and non-chaotic behavior. The new characterization algorithm is successfully applied to irregularly sampled data generated by a model as well as data consisting of observations of variable stars.

There exist many algorithms for detecting chaos in time series data. In order to successfully detect chaos, those algorithms require regular sampling rates. Because real-world data are often irregularly sampled, detecting chaos using real-world data can be difficult or impossible with pre-existing algorithms. This paper presents a characterization method which uses the Lomb-Scargle Periodogram (LSP) to compute the power spectrum of an irregularly sampled time series. The power spectrum can then be analyzed to detect chaos in the time series. The method is shown to successfully characterize model time series and time series observed from variable stars. The work presented here is a modification of the work done by Wiebe and Virgin.

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The author would like to thank Professor E. R. Tracy of The College of William and Mary for his helpful comments on this manuscript. The author would also like to thank the reviewers for their helpful comments and suggestions.

I. INTRODUCTION

II. THE CHARACTERIZATION ALGORITHM

A. The heuristic method

B. The modified heuristic method

III. APPLICATION

A. The Lorenz equations

B. Variable star data

IV. CONCLUSION

### Key Topics

- Time series analysis
- 56.0
- Chaos
- 24.0
- Variable stars
- 17.0
- Density functional theory
- 13.0
- Chaotic dynamics
- 9.0

## Figures

Part of the time series generated from (1) using (a) for chaotic dynamics and (b) for periodic dynamics.

Part of the time series generated from (1) using (a) for chaotic dynamics and (b) for periodic dynamics.

The power spectrum and point count plot for (1) using ((a) and (c)) for chaotic dynamics and ((b) and (d)) for periodic dynamics. Note that Figure 2(b) contains the actual computed points in the spectrum, those points are not shown in Figure 2(a) as they make that figure difficult to read.

The LSP power spectrum and point count plot for (1) using ((a) and (c)) for chaotic dynamics and ((b) and (d)) for periodic dynamics. Note that Figure 3(b) contains the actual computed points in the spectrum, those points are not shown in Figure 3(a) as they make that figure difficult to read.

The LSP power spectrum and point count plot for (1) using ((a) and (c)) for chaotic dynamics and ((b) and (d)) for periodic dynamics. Note that Figure 3(b) contains the actual computed points in the spectrum, those points are not shown in Figure 3(a) as they make that figure difficult to read.

The irregularly sampled time series generated from (1) . Each graph is labeled by the type of dynamics and the percentage of data removed.

The irregularly sampled time series generated from (1) . Each graph is labeled by the type of dynamics and the percentage of data removed.

The results of the heuristic method for the irregularly sampled chaotic time series generated from (1) . The left column contains the power spectrum as computed by the DFT and the right column contains the point count plot. Each graph is labeled by the type of dynamics and the percentage of data removed.

The results of the heuristic method for the irregularly sampled chaotic time series generated from (1) . The left column contains the power spectrum as computed by the DFT and the right column contains the point count plot. Each graph is labeled by the type of dynamics and the percentage of data removed.

The results of the heuristic method for the irregularly sampled periodic time series generated from (1) . The left column contains the power spectrum as computed by the DFT and the right column contains the point count plot. Each graph is labelled by the type of dynamics and the percentage of data removed.

The results of the heuristic method for the irregularly sampled periodic time series generated from (1) . The left column contains the power spectrum as computed by the DFT and the right column contains the point count plot. Each graph is labelled by the type of dynamics and the percentage of data removed.

The results of the modified heuristic method for the irregularly sampled chaotic time series generated from (1) . The left column contains the power spectrum as computed by the LSP and the right column contains the point count plot. Each graph is labelled by the type of dynamics and the percentage of data removed.

The results of the modified heuristic method for the irregularly sampled chaotic time series generated from (1) . The left column contains the power spectrum as computed by the LSP and the right column contains the point count plot. Each graph is labelled by the type of dynamics and the percentage of data removed.

The results of the modified heuristic method for the irregularly sampled periodic time series generated from (1) . The left column contains the power spectrum as computed by the LSP and the right column contains the point count plot. Each graph is labelled by the type of dynamics and the percentage of data removed.

The results of the modified heuristic method for the irregularly sampled periodic time series generated from (1) . The left column contains the power spectrum as computed by the LSP and the right column contains the point count plot. Each graph is labelled by the type of dynamics and the percentage of data removed.

The time series data (top), normalized power spectrum (middle), and point count plot (bottom) for Car.

The time series data (top), normalized power spectrum (middle), and point count plot (bottom) for Car.

The time series data (top), normalized power spectrum (middle), and point count plot (bottom) for V CVn.

The time series data (top), normalized power spectrum (middle), and point count plot (bottom) for V CVn.

The time series data (top), normalized power spectrum (middle), and point count plot (bottom) for UX Dra.

The time series data (top), normalized power spectrum (middle), and point count plot (bottom) for UX Dra.

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