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Feigenbaum Graphs: A Complex Network Perspective of Chaos

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  • Bartolo Luque
  • Lucas Lacasa
  • Fernando J Ballesteros
  • Alberto Robledo

Abstract

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.

Suggested Citation

  • Bartolo Luque & Lucas Lacasa & Fernando J Ballesteros & Alberto Robledo, 2011. "Feigenbaum Graphs: A Complex Network Perspective of Chaos," PLOS ONE, Public Library of Science, vol. 6(9), pages 1-8, September.
    • Handle: RePEc:plo:pone00:0022411
    DOI: 10.1371/journal.pone.0022411
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    References listed on IDEAS

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    1. Steven H. Strogatz, 2001. "Exploring complex networks," Nature, Nature, vol. 410(6825), pages 268-276, March.
    2. Gutin, Gregory & Mansour, Toufik & Severini, Simone, 2011. "A characterization of horizontal visibility graphs and combinatorics on words," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(12), pages 2421-2428.
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    Cited by:

    1. Jorge Calero-Sanz, 2022. "On the Degree Distribution of Haros Graphs," Mathematics, MDPI, vol. 11(1), pages 1-15, December.
    2. Bezsudnov, I.V. & Snarskii, A.A., 2014. "From the time series to the complex networks: The parametric natural visibility graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 53-60.
    3. Nakamura, Tomomichi & Tanizawa, Toshihiro, 2012. "Networks with time structure from time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4704-4710.
    4. Wang, Fan & Tian, Lixin & Du, Ruijin & Dong, Gaogao, 2021. "Universal law in the crude oil market based on visibility graph algorithm and network structure," Resources Policy, Elsevier, vol. 70(C).
    5. Robert G. Sacco, 2019. "The Predictability of Synchronicity Experience: Results from a Survey of Jungian Analysts," International Journal of Psychological Studies, Canadian Center of Science and Education, vol. 11(3), pages 1-46, September.
    6. Davide Provenzano & Rodolfo Baggio, 2021. "Complexity traits and synchrony of cryptocurrencies price dynamics," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(2), pages 941-955, December.
    7. Wu, Zhenyu & Shang, Pengjian & Xiong, Hui, 2018. "An improvement of the measurement of time series irreversibility with visibility graph approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 370-378.
    8. de Godoy, Isabelle Bueno Silva & McGrane-Corrigan, Blake & Mason, Oliver & Moral, Rafael de Andrade & Godoy, Wesley Augusto Conde, 2023. "Plant-host shift, spatial persistence, and the viability of an invasive insect population," Ecological Modelling, Elsevier, vol. 475(C).
    9. Nuño, Juan Carlos & Muñoz, Francisco J., 2020. "The partial visibility curve of the Feigenbaum cascade to chaos," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    10. Gao, Meng & Ge, Ruijun, 2024. "Mapping time series into signed networks via horizontal visibility graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 633(C).

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