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Chaos / Volume 22 / Issue 1 / REGULAR ARTICLES

Chaos 22, 013109 (2012); http://dx.doi.org/10.1063/1.3676686 (14 pages)

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario

Bartolo Luque1, Lucas Lacasa1, Fernando J. Ballesteros2, and Alberto Robledo3

1Departamento de Matemática Aplicada y Estadística, ETSI Aeronáuticos, Universidad Politécnica de Madrid, Spain
2Observatori Astronòmic, Universitat de València, Spain
3Instituto de Física y Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Mexico

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(Received 12 September 2011; accepted 22 December 2011; published online 24 January 2012)

Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [B. Luque et al., PLoS ONE 6, 9 (2011)] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here, we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree distributions, mean distances, clustering coefficients, etc., associated to the bifurcation cascades and their accumulation points. We describe how the resultant families of graphs can be framed into a renormalization group scheme in which fixed-point graphs reveal their scaling properties. These fixed points are then re-derived from an entropy optimization process defined for the graph sets, confirming a suggested connection between renormalization group and entropy optimization. Finally, we provide analytical and numerical results for the graph entropy and show that it emulates the Lyapunov exponent of the map independently of its sign.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. FEIGENBAUM GRAPHS
    1. Mean degree math
    2. Normalized mean distance math
  3. PERIOD-DOUBLING ROUTE TO CHAOS: RESULTS
    1. Order of visits of stable branches and chaotic bands
    2. Topological properties of Feigenbaum graphs along the period-doubling cascade
      1. Degree distribution P(n,k)
      2. Mean degree math(n) and normalized distance math(n)
      3. Clustering coefficient c(n, k)
      4. Higher moments of the degree distribution: Variance σ2(n)
  4. REVERSE BIFURCATION CASCADE OF CHAOTIC BANDS: RESULTS
    1. Self-affine properties of chaotic bands: Mean degree and degree distribution
    2. Periodic windows: Self-affine copies of the Feingenbaum diagram
  5. RENORMALIZATION GROUP APPROACH
    1. RG transformation: Definition, flows, and fixed points
    2. Crossover phenomenon
  6. NETWORK ENTROPY
    1. Entropy optimization and RG fixed points
    2. Network entropy and Pesin theorem
      1. Periodic attractors
      2. Chaotic attractors
  7. SUMMARY

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KEYWORDS and PACS

Keywords

bifurcation, entropy, Lyapunov methods, nonlinear dynamical systems, numerical analysis, optimisation, renormalisation, time series

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3676686

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    B. Luque, L. Lacasa, F. Ballesteros, and J. Luque, Phys. Rev. E 80, 046103 (2009).

    S. A. Marvel, R. E. Mirollo, and S. Strogatz, Chaos 19, 043104 (2009)CHAOEH000019000004043104000001.

    F. Radicchi, J. J. Ramasco, A. Barrat, and S. Fortunato, Phys. Rev. Lett. 101, 148701 (2008).

    A. Robledo, Phys. Rev. Lett. 83, 12 (1999).

    E. T. Jaynes, Phys. Rev. 106, 4 (1957).


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