Skip to main content
SpringerLink
Log in

The geometry of chaotic dynamics — a complex network perspective

  • Regular Article
  • Focus Section on Frontiers in Network Science: Advances and Applications
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ε-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ε-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ε-recurrence networks exhibit an important link between dynamical systems and graph theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Phys. Rep. 438, 237 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  2. J.P. Eckmann, S.O. Kamphorst, D. Ruelle, Europhys. Lett. 4, 973 (1987)

    Article  ADS  Google Scholar 

  3. N. Marwan, Eur. Phys. J. ST 164, 3 (2008)

    Google Scholar 

  4. H. Poincaré, Acta Mathematica 13, A3 (1890)

    Article  Google Scholar 

  5. M. Thiel, M.C. Romano, J. Kurths, Phys. Lett. A 330, 343 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Y. Hirata, S. Horai, K. Aihara, Eur. Phys. J. ST 164, 13 (2008)

    Google Scholar 

  7. G. Robinson, M. Thiel, Chaos 19, 023104 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  8. M. Thiel, M.C. Romano, P.L. Read, J. Kurths, Chaos 14, 234 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. R. Albert, A.L. Barabasi, Rev. Mod. Phys. 74, 47 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. M.E.J. Newman, SIAM Rev. 45, 167 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U. Hwang, Phys. Rep. 424, 175 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  12. L.D.F. Costa, F.A. Rodrigues, G. Travieso, P.R.V. Boas, Adv. Phys. 56, 167 (2007)

    Article  ADS  Google Scholar 

  13. N. Marwan, J.F. Donges, Y. Zou, R.V. Donner, J. Kurths, Phys. Lett. A 373, 4246 (2009)

    Article  ADS  Google Scholar 

  14. Z. Gao, N. Jin, Phys. Rev. E 79, 066303 (2009a)

    Article  ADS  Google Scholar 

  15. Z. Gao, N. Jin, Chaos 19, 033137 (2009b)

    Article  ADS  Google Scholar 

  16. R.V. Donner, Y. Zou, J.F. Donges, N. Marwan, J. Kurths, Phys. Rev. E 81, R015101 (2010)

    Article  ADS  Google Scholar 

  17. R.V. Donner, Y. Zou, J.F. Donges, N. Marwan, J. Kurths, New J. Phys. 12, 033025 (2010)

    Article  ADS  Google Scholar 

  18. R.V. Donner, M. Small, J.F. Donges, N. Marwan, Y. Zou, R. Xiang, J. Kurths, Int. J. Bifurc. Chaos (in press), arXiv:1010.6032

  19. M. Penrose, Random Geometric Graphs (Oxford University Press, Oxford, 2003)

  20. S. Felsner, Geometric Graphs and Arrangements, 3rd edn. (Vieweg, Wiesbaden, 2004)

  21. C. Herrmann, M. Barthélemy, P. Provero, Phys. Rev. E 68, 026128 (2003)

    Article  ADS  Google Scholar 

  22. M.A. Carreira-Perpiñan, R.S. Zemel, Proximity graphs for clustering and manifold learning, in Advances in Neural Information Processing Systems 17 (NIPS 2004), edited by L.K. Saul, Y. Weiss, L. Bottou (MIT Press, Cambridge, 2005), pp. 225–232

  23. C. Zhou, L. Zemanova, G. Zamora, C.C. Hilgetag, J. Kurths, Phys. Rev. Lett. 97, 238103 (2006)

    Article  ADS  Google Scholar 

  24. C. Zhou, L. Zemanova, G. Zamora-Lopez, C.C. Hilgetag, J. Kurths, New J. Phys. 9, 178 (2007)

    Article  ADS  Google Scholar 

  25. J.F. Donges, Y. Zou, N. Marwan, J. Kurths, Eur. Phys. J. ST 174, 157 (2009)

    Google Scholar 

  26. J.F. Donges, Y. Zou, N. Marwan, J. Kurths, Europhys. Lett. 87, 48007 (2009)

    Article  ADS  Google Scholar 

  27. I. Borg, P. Groenen, Modern Multidimensional Scaling: theory and applications, 2nd edn. (Springer, New York, 2005)

  28. J.B. Tenenbaum, V. de Silva, J.C. Langford, Science 290, 2319 (2000)

    Article  ADS  Google Scholar 

  29. M. Dellnitz, M. Hessel-von Molo, P. Metzner, R. Preis, C. Schütte, in Analysis, Modeling and Simulation of Multiscale Problems, edited by A. Mielke (Springer, Heidelberg, 2006), pp. 619–646

  30. K. Padberg, B. Thiere, R. Preis, M. Dellnitz, Communications in Nonlinear Science and Numerical Simulation 14, 4176 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. G. Nicolis, A. García Cantú, C. Nicolis, Int. J. Bifurc. Chaos 15, 3467 (2005)

    Article  MATH  Google Scholar 

  32. J. Zhang, M. Small, Phys. Rev. Lett. 96, 238701 (2006)

    Article  ADS  Google Scholar 

  33. Y. Yang, H. Yang, Physica A 387, 1381 (2008)

    Article  ADS  Google Scholar 

  34. L. Lacasa, B. Luque, F. Ballesteros, J. Luque, J.C. Nuno, Proceedings of the National Academy of Sciences USA 105, 4972 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. Y. Shimada, T. Kimura, T. Ikeguchi, Analysis of Chaotic Dynamics Using Measures of the Complex Network Theory, in Artificial Neural Networks - ICANN 2008, Pt. I, edited by V. Kurkova, R. Neruda, J. Koutnik Lecture Notes in Computer Science (Springer, New York, 2008), Vol. 5163, pp. 61–70

  36. X. Xu, J. Zhang, M. Small, Proceedings of the National Academy of Sciences USA 105, 19601 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Y. Zou, R.V. Donner, J.F. Donges, N. Marwan, J. Kurths, Chaos 20, 043130 (2010)

    Article  ADS  Google Scholar 

  38. A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Phys. Rep. 469, 93 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  39. S.V. Buldyrev, R. Parshani, G. Paul, H.E. Stanley, S. Havlin, Nature 464, 1025 (2010)

    Article  ADS  Google Scholar 

  40. J.C. Oxtoby, Proceedings of the National Academy of Sciences USA 23, 443 (1937)

    Article  ADS  Google Scholar 

  41. A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems (Cambridge University Press, Cambridge, 1995)

  42. D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)

    Article  ADS  Google Scholar 

  43. A. Barrat, M. Weigt, Eur. Phys. J. B 13, 547 (2000)

    Article  ADS  Google Scholar 

  44. M.E.J. Newman, Phys. Rev. E 64, 016131 (2001)

    Article  ADS  Google Scholar 

  45. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 65, 066122 (2002)

    Article  ADS  Google Scholar 

  46. G. Szabó, M. Alava, J. Kertész, Phys. Rev. E 67, 056102 (2003)

    Article  ADS  Google Scholar 

  47. E. Ravasz, A.L. Somera, D.A. Mongru, Z.N. Oltvai, A.L. Barabasi, Science 297, 1551 (2002)

    Article  ADS  Google Scholar 

  48. E. Ravasz, A.L. Barabási, Phys. Rev. E 67, 026112 (2003)

    Article  ADS  Google Scholar 

  49. A. Vázquez, Phys. Rev. E 67, 056104 (2003)

    Article  ADS  Google Scholar 

  50. P. Grassberger, Phys. Lett. A 97, 227 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  51. P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50, 346 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  52. J.G. Reid, T.A. Trainor (2003), arXiv:math-ph/0305022

  53. J.D. Farmer, E. Ott, J.A. Yorke, Physica D 7, 153 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  54. J. Kaplan, J. Yorke, in Functional Differential Equations and Approximation of Fixed Points, edited by H.O. Peitgen, H.O. Walther, Lecture Notes in Mathematics (Springer Berlin/Heidelberg, 1979), Vol. 730, pp. 204–227

  55. E. Ott, Chaos in Dynamical Systems, 2nd edn. (Cambridge University Press, Cambridge, 2002)

  56. B.R. Hunt, Nonlinearity 9, 845 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  57. K. Gelfert, Journal for Analysis and its Applications 22, 553 (2003)

    MATH  MathSciNet  Google Scholar 

  58. Y. Zou, J. Heitzig, J.D. Farmer, R. Meucci, S. Euzzor, N. Marwan, R.V. Donner, J.F. Donges, J. Kurths (in prep.)

  59. L. Lacasa, B. Luque, J. Luque, J.C. Nuno, Europhys. Lett. 86, 30001 (2009)

    Article  ADS  Google Scholar 

  60. X.H. Ni, Z.Q. Jiang, W.X. Zhou, Phys. Lett. A 373, 3822 (2009)

    Article  ADS  Google Scholar 

  61. J. Dall, M. Christensen, Phys. Rev. E 66, 016121 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  62. J.C. Sprott, Chaos and Time-Series Analysis (Oxford University Press, Oxford, 2003)

  63. J. Heitzig, J.F. Donges, Y. Zou, N. Marwan, J. Kurths (2011), arXiv:1101.4757 [physics.data-an]

  64. E.M. Oblow, Phys. Lett. A 128, 406 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  65. N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, J. Kurths, Phys. Rev. E 66, 026702 (2002)

    Article  ADS  Google Scholar 

  66. R.V. Donner, J.F. Donges, Y. Zou, N. Marwan, J. Kurths, Proc. NOLTA 2010 (2010), pp. 87–90

  67. A. Veronig, M. Messerotti, A. Hanslmeier, A&A 357, 337 (2000)

    ADS  Google Scholar 

  68. S. Gratrix, J.N. Elgin, Phys. Rev. Lett. 92, 014101 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  69. P. Grassberger, I. Procaccia, Physica D 9, 189 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  70. D.P. Lathrop, E.J. Kostelich, Phys. Rev. A 40, 4028 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  71. C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. A 37, 1711 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  72. M. Hénon, Commun. Math. Phys. 50, 69 (1976)

    Article  MATH  ADS  Google Scholar 

  73. P. Cvitanović, G.H. Gunaratne, I. Procaccia, Phys. Rev. A 38, 1503 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  74. J.A.C. Gallas, Phys. Rev. Lett. 70, 2714 (1993)

    Article  ADS  Google Scholar 

  75. J.A.C. Gallas, Physica A 202, 196 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  76. M. Thiel, Ph.D. thesis, University of Potsdam, 2004

  77. C. Bonatto, J.A.C. Gallas, Phil. Trans. R. Soc. A 366, 505 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  78. J.A.C. Gallas, Int. J. Bifurc. Chaos 20, 197 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  79. Y. Saiki, Nonlinear Processes in Geophysics 14, 615 (2007)

    Article  ADS  Google Scholar 

  80. G. Csárdi, T. Nepusz, InterJournal CX.18, 1695 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412, Potsdam, Germany

    R. V. Donner, J. Heitzig, J. F. Donges, Y. Zou, N. Marwan & J. Kurths

  2. Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489, Berlin, Germany

    J. F. Donges & J. Kurths

Authors
  1. R. V. Donner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Heitzig
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. F. Donges
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. N. Marwan
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. J. Kurths
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. V. Donner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Donner, R.V., Heitzig, J., Donges, J.F. et al. The geometry of chaotic dynamics — a complex network perspective. Eur. Phys. J. B 84, 653–672 (2011). https://doi.org/10.1140/epjb/e2011-10899-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2011-10899-1

Keywords

Search

Navigation