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  • Open Access

Quantifying the irregularity of a time series

Max Potratzki1,2, Manuel Adams1,2, Timo Bröhl1,2, and Klaus Lehnertz1,2,3,*

  • 1Department of Epileptology, University of Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany
  • 2Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14–16, 53115 Bonn, Germany
  • 3Interdisciplinary Center for Complex Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany

  • *Contact author: klaus.lehnertz@ukbonn.de
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Phys. Rev. Research 8, 013001 – Published 2 January, 2026

DOI: https://doi.org/10.1103/3vzd-7kg2

Abstract

We introduce circulance, a scalar measure for classifying time series of dynamical systems. Circulance captures the extent of temporal regularity or irregularity that is encoded in the topology of a directed ordinal pattern transition network derived from a time series. We demonstrate numerically that circulance sensitively and robustly positions time series of canonical model systems, representative of preset dynamical regimes, along a continuous spectrum from regularity to randomness. Analyzing empirical data from long-term observations of high-dimensional, complex systems—human brain and the Sun—reveals that circulance aids in elucidating different dynamical regimes.

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References (84)

  1. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. (Cambridge University Press, Cambridge, UK, 2003).
  2. J. Sun, Y. Zhao, T. Nakamura, and M. Small, From phase space to frequency domain: A time-frequency analysis for chaotic time series, Phys. Rev. E 76, 016220 (2007).
  3. L. Lacasa, B. Luque, F. Ballesteros, J. Luque, and J. C. Nuno, From time series to complex networks: The visibility graph, Proc. Natl. Acad. Sci. USA 105, 4972 (2008).
  4. L. Lacasa and R. Toral, Description of stochastic and chaotic series using visibility graphs, Phys. Rev. E 82, 036120 (2010).
  5. P. Bloomfield, Fourier Analysis of Time Series: An Introduction (John Wiley & Sons, New York, 2004).
  6. G. A. Gottwald and I. Melbourne, A new test for chaos in deterministic systems, Proc. R. Soc. Lond. A 460, 603 (2004).
  7. N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, Recurrence plots for the analysis of complex systems, Phys. Rep. 438, 237 (2007).
  8. U. S. Freitas, C. Letellier, and L. A. Aguirre, Failure in distinguishing colored noise from chaos using the “noise titration” technique, Phys. Rev. E 79, 035201(R) (2009).
  9. C. W. Kulp and S. Smith, Characterization of noisy symbolic time series, Phys. Rev. E 83, 026201 (2011).
  10. C. Kulp and L. Zunino, Discriminating chaotic and stochastic dynamics through the permutation spectrum test, Chaos 24, 033116 (2014).
  11. D. M. Mateos, L. E. Riveaud, and P. W. Lamberti, Detecting dynamical changes in time series by using the Jensen Shannon divergence, Chaos 27, 083118 (2017).
  12. Y. Zou, R. V. Donner, N. Marwan, J. F. Donges, and J. Kurths, Complex network approaches to nonlinear time series analysis, Phys. Rep. 787, 1 (2019).
  13. M. R. R. Tabar, Analysis and Data-based Reconstruction of Complex Nonlinear Dynamical Systems: Using the Methods of Stochastic Processes (Springer, Cham, Switzerland, 2019).
  14. J. R. Tempelman and F. A. Khasawneh, A look into chaos detection through topological data analysis, Phys. D: Nonlinear Phenom. 406, 132446 (2020).
  15. J. D. Hamilton, Time Series Analysis (Princeton University Press, Princeton, NJ, 2020).
  16. I. Kottlarz and U. Parlitz, Ordinal pattern-based complexity analysis of high-dimensional chaotic time series, Chaos 33, 053105 (2023).
  17. A. K. Angelidis, K. Goulas, C. Bratsas, G. C. Makris, M. P. Hanias, S. G. Stavrinides, and I. E. Antoniou, Distinction of chaos from randomness is not possible from the degree distribution of the visibility and phase space reconstruction graphs, Entropy 26, 341 (2024).
  18. M. Small, D. Yu, and R. G. Harrison, Surrogate test for pseudoperiodic time series data, Phys. Rev. Lett. 87, 188101 (2001).
  19. X. Luo, T. Nakamura, and M. Small, Surrogate test to distinguish between chaotic and pseudoperiodic time series, Phys. Rev. E 71, 026230 (2005).
  20. Y. Zou, D. Pazó, M. Romano, M. C. Thiel, and J. Kurths, Distinguishing quasiperiodic dynamics from chaos in short-time series, Phys. Rev. E 76, 016210 (2007).
  21. O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99, 154102 (2007).
  22. Z. Gao and N. Jin, Complex network from time series based on phase space reconstruction, Chaos 19, 033137 (2009).
  23. B. Luque, L. Lacasa, F. Ballesteros, and J. Luque, Horizontal visibility graphs: Exact results for random time series, Phys. Rev. E 80, 046103 (2009).
  24. N. Marwan, J. F. Donges, Y. Zou, R. V. Donner, and J. Kurths, Complex network approach for recurrence analysis of time series, Phys. Lett. A 373, 4246 (2009).
  25. Y. Zou, R. V. Donner, J. F. Donges, N. Marwan, and J. Kurths, Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods, Chaos 20, 043130 (2010).
  26. E. J. Ngamga, D. V. Senthilkumar, A. Prasad, P. Parmananda, N. Marwan, and J. Kurths, Distinguishing dynamics using recurrence-time statistics, Phys. Rev. E 85, 026217 (2012).
  27. S. Das, C. B. Dock, Y. Saiki, M. Salgado-Flores, E. Sander, J. Wu, and J. A. Yorke, Measuring quasiperiodicity, Europhys. Lett. 114, 40005 (2016).
  28. F. Olivares, M. Zanin, L. Zunino, and D. G. Pérez, Contrasting chaotic with stochastic dynamics via ordinal transition networks, Chaos 30, 063101 (2020).
  29. Z. Zhang, M. Zhang, Y. Chen, Z. Xiang, J. Xu, and X. Zhou, Distinguishing chaotic from stochastic dynamics via the complexity of ordinal patterns, AIP Adv. 11, 045122 (2021).
  30. B. Boaretto, R. C. Budzinski, K. L. Rossi, T. L. Prado, S. R. Lopes, and C. Masoller, Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks, Sci. Rep. 11, 15789 (2021).
  31. T. L. Prado, B. R. R. Boaretto, G. Corso, G. Z. dos Santos Lima, J. Kurths, and S. R. Lopes, A direct method to detect deterministic and stochastic properties of data, New J. Phys. 24, 033027 (2022).
  32. W. Ren and Z. Jin, Phase space visibility graph, Chaos Solitons Fract. 176, 114170 (2023).
  33. J. S. A. E. Fouda, W. Koepf, N. Marwan, J. Kurths, and T. Penzel, Complexity from ordinal pattern positioned slopes (copps), Chaos Solitons Fract. 181, 114708 (2024).
  34. A. Mohan, G. Ambika, and C. Meena, Deep learning for classifying dynamical states from time series via recurrence plots, arXiv:2506.17498.
  35. M. B. Priestley, Nonlinear and Non-stationary Time Series Analysis (Academic Press, London, 1988).
  36. R. Manuca and R. Savit, Stationarity and nonstationarity in time series analysis, Phys. D: Nonlinear Phenom. 99, 134 (1996).
  37. K. Fokianos and B. Kedem, Prediction and classification of non-stationary categorical time series, J. Multivar. Anal. 67, 277 (1998).
  38. R. Hegger, H. Kantz, L. Matassini, and T. Schreiber, Coping with non-stationarity by overembedding, Phys. Rev. Lett. 84, 4092 (2000).
  39. C. Rieke, K. Sternickel, R. G. Andrzejak, C. E. Elger, P. David, and K. Lehnertz, Measuring nonstationarity by analyzing the loss of recurrence in dynamical systems, Phys. Rev. Lett. 88, 244102 (2002).
  40. P. Fryzlewicz and H. Ombao, Consistent classification of nonstationary time series using stochastic wavelet representations, J. Am. Stat. Assoc. 104, 299 (2009).
  41. K. Lepage and D. Thomson, Spectral analysis of cyclostationary time-series: A robust method, Geophys. J. Int. 179, 1199 (2009).
  42. C. Bandt and B. Pompe, Permutation entropy—A complexity measure for time series, Phys. Rev. Lett. 88, 174102 (2002).
  43. A. M. Unakafov and K. Keller, Conditional entropy of ordinal patterns, Phys. D: Nonlinear Phenom. 269, 94 (2014).
  44. J. Zhang, J. Zhou, M. Tang, H. Guo, M. Small, and Y. Zou, Constructing ordinal partition transition networks from multivariate time series, Sci. Rep. 7, 7795 (2017).
  45. D. Cuesta-Frau, A. Molina-Picó, B. Vargas, and P. González, Permutation entropy: Enhancing discriminating power by using relative frequencies vector of ordinal patterns instead of their shannon entropy, Entropy 21, 1013 (2019).
  46. M. Zanin and F. Olivares, Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series, Commun. Phys. 4, 190 (2021).
  47. M. Small, Complex networks from time series: Capturing dynamics, in 2013 IEEE International Symposium on Circuits and Systems (ISCAS) (IEEE, Piscataway, NJ, 2013), pp. 2509–2512.
  48. M. McCullough, M. Small, T. Stemler, and H. H.-C. Iu, Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems, Chaos 25, 053101 (2015).
  49. A. A. Pessa and H. V. Ribeiro, Characterizing stochastic time series with ordinal networks, Phys. Rev. E 100, 042304 (2019).
  50. M. Seeck, L. Koessler, T. Bast, F. Leijten, C. Michel, C. Baumgartner, B. He, and S. Beniczky, The standardized EEG electrode array of the IFCN, Clin. Neurophysiol. 128, 2070 (2017).
  51. F. Clette, L. Svalgaard, J. M. Vaquero, and E. W. Cliver, Revisiting the sunspot number: A 400-year perspective on the solar cycle, Space Sci. Rev. 186, 35 (2014).
  52. F. Clette and L. Lefèvre, Silso Sunspot Number v2.0 (WDC SILSO, Royal Observatory of Belgium (ROB), 2015), https://doi.org/10.24414/qnza-ac80.
  53. In some distinct and rare cases within periodic regimes of the Hénon map, the respective circulance value can be zero.
  54. Estimating circulance for a time series of 104 data points can be performed in few milliseconds on a PC with a CPU clock rate of 3.7 GHz.
  55. M. Hénon, A two-dimensonal mapping with a strange attractor, Commun. Math. Phys. 50, 69 (1976).
  56. U. Parlitz, Identification of true and spurious Lyapunov exponents from time series, Int. J. Bifurc. Chaos 02, 155 (1992).
  57. R. G. Andrzejak, G. Widman, K. Lehnertz, P. David, and C. E. Elger, The epileptic process as nonlinear deterministic dynamics in a stochastic environment: An evaluation on mesial temporal lobe epilepsy, Epilepsy Res. 44, 129 (2001).
  58. J. P. Pijn, J. van Neerven, A. Noest, and F. H. L. da Silva, Chaos or noise in EEG signals: Dependence of state and brain site, Electroenceph. Clin. Neurophysiol. 79, 371 (1991).
  59. D. Prichard and J. Theiler, Generating surrogate data for time series with several simultaneously measured variables, Phys. Rev. Lett. 73, 951 (1994).
  60. C. J. Stam, Nonlinear dynamical analysis of EEG and MEG: Review of an emerging field, Clin. Neurophysiol. 116, 2266 (2005).
  61. S. Tong and N. V. Thankor, Quantitative EEG Analysis Methods and Clinical Applications (Artech House, Norwood, 2009).
  62. K. Lehnertz, T. Rings, and T. Bröhl, Time in brain: How biological rhythms impact on EEG signals and on EEG-derived brain networks, Front. Networks Physiol. 1, 755016 (2021).
  63. T. Bröhl, R. Von Wrede, and K. Lehnertz, Impact of biological rhythms on the importance hierarchy of constituents in time-dependent functional brain networks, Front. Networks Physiol. 3, 1237004 (2023).
  64. K. Lehnertz, J. Arnhold, P. Grassberger, and C. E. Elger, Chaos in Brain? (World Scientific, Singapore, 2000).
  65. H. Korn and P. Faure, Is there chaos in the brain? II. Experimental evidence and related models, C. R. Biol. 326, 787 (2003).
  66. K. Lehnertz, C. Geier, T. Rings, and K. Stahn, Capturing time-varying brain dynamics, EPJ Nonlinear Biomed. Phys. 5, 2 (2017).
  67. D. Aeschbach and A. A. Borbely, All-night dynamics of the human sleep EEG, J. Sleep Res. 2, 70 (1993).
  68. M. A. Carskadon and W. C. Dement, Normal human sleep: An overview, in Principles and Practice of Sleep Medicine, edited by M. H. Kryger, T. Roth, and W. C. Dement, 5th ed. (W.B. Saunders, Philadelphia, 2011), Chap. 2, pp. 16–26.
  69. S. Zhou, Y. Feng, W.-Y. Wu, Y. Li, and J. Liu, Low-dimensional chaos and fractal properties of long-term sunspot activity, Res. Astron. Astrophys. 14, 104 (2014).
  70. L. Deng, B. Li, Y. Xiang, and G. Dun, Comparison of chaotic and fractal properties of polar faculae with sunspot activity, Astron. J. 151, 2 (2016).
  71. C. Letellier, L. Aguirre, J. Maquet, and R. Gilmore, Evidence for low dimensional chaos in sunspot cycles, Astron. Astrophys. 449, 379 (2006).
  72. D. H. Hathaway, The solar cycle, Living Rev. Sol. Phys. 12, 4 (2015).
  73. G. de Toma, O. R. White, G. A. Chapman, S. R. Walton, D. G. Preminger, and A. M. Cookson, Solar cycle 23: An anomalous cycle? Astrophys. J. 609, 1140 (2004).
  74. S. Vashishtha and K. R. Sreenivasan, Bistability in the sunspot cycle, Europhys. Lett. 148, 23001 (2024).
  75. N. Jevtić, J. Schweitzer, and C. Cellucci, Research note: Nonlinear time series analysis of northern and southern solar hemisphere daily sunspot numbers in search of short-term chaotic behavior, Astron. Astrophys. 379, 611 (2001).
  76. B. Van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev. 1, 701 (1920).
  77. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445 (1961).
  78. G. Ansmann, R. Karnatak, K. Lehnertz, and U. Feudel, Extreme events in excitable systems and mechanisms of their generation, Phys. Rev. E 88, 052911 (2013).
  79. G. Baier and M. Klein, Maximum hyperchaos in generalized hénon maps, Phys. Lett. A 151, 281 (1990).
  80. J. Vano, J. Wildenberg, M. Anderson, J. Noel, and J. Sprott, Chaos in low-dimensional Lotka–Volterra models of competition, Nonlinearity 19, 2391 (2006).
  81. G. M. Zaslavsky, The simplest case of a strange attractor, Phys. Lett. A 69, 145 (1978).
  82. E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20, 130 (1963).
  83. E. N. Lorenz, Predictability: A problem partly solved, in Proceedings of Seminar on Predictability, edited by T. Palmer and R. Hagedorn (Cambridge University Press, 1996), Vol. 1, pp. 1–18.
  84. T. Bröhl and M. Potratzki, OPyTN: Ordinal pattern transition networks in Python, https://github.com/timocbro/OPyTN (2025).

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Phys. Rev. Research 8, 013001– Published 2 January, 2026
  • Received 4 July 2025
  • Accepted 5 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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