Abstract
A novel class of graphs, here named quasiperiodic, are constructed via application of the Horizontal Visibility algorithm to the time series generated along the quasiperiodic route to chaos. We show how the hierarchy of mode-locked regions represented by the Farey tree is inherited by their associated graphs. We are able to establish, via Renormalization Group (RG) theory, the architecture of the quasiperiodic graphs produced by irrational winding numbers with pure periodic continued fraction. Finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.
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Campanharo, A.S.L.O., Sirer, M.I., Malmgren, R.D., Ramos, F.M., Amaral, L.A.N.: Duality between time series and networks. PLoS ONE 6, e23378 (2011)
Donner, R.V., Zou, Y., Donges, J.F., Marwan, N., Kurths, J.: Recurrence networks a novel paradigm for nonlinear time series analysis. New J. Phys. 12, 033025 (2010a)
Donner, R.V., et al.: Recurrence-based time series analysis by means of complex network methods. Int. J. Bifurct. Chaos 21, 1019 (2010b)
Donner, R.V., et al.: The geometry of chaotic dynamics. A complex network perspective. Eur. Phys. J. B 84, 653 (2011)
Hao, B.-H., Zeng, W.-M.: Applied symbolic dynamics and chaos. World Scientific, Singapore (1998)
Hilborn, R.C.: Chaos and Nonlinear Dynamics. Oxford University Press, New York (1994)
Kyriakopoulos, F., Thurner, S.: Directed network representation of discrete dynamical maps. Lect. Notes Comput. Sci. 4488, 625–632 (2007)
Lacasa, L., Luque, B., Ballesteros, F., Luque, J., Nuño, J.C.: From time series to complex networks: the visibility graph. Proc. Natl. Acad. Sci. USA 105, 4972 (2008)
Luque, B., Lacasa, L., Ballesteros, F., Luque, J.: Horizontal visibility graphs: exact results for random time series. Phys. Rev. E 80, 046103 (2009)
Luque, B., Lacasa, L., Ballesteros, F.J., Robledo, A.: Feigenbaum graphs: a complex network perspective of chaos. PLoS ONE 6, e22411 (2011)
Luque, B., Lacasa, L., Ballesteros, F.J., Robledo, A.: Analytical properties of horizontal visibility graphs in the Feigenbaum scenario. Chaos 22, 013109 (2012)
Schroeder, M.: Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Freeman, New York (1991)
Schuster, H.G.: Deterministic Chaos. An Introduction, 2nd revised edn. VCH, Weinheim (1988)
Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic Phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951 (1984)
Strogatz, S.H.: Nonlinear Dynamics and Chaos. Perseus Books Publishing, LLC, Reading (1994)
Xu, X., Zhang, J., Small, M.: Superfamily phenomena and motifs of networks induced from time series. Proc. Natl. Acad. Sci. USA 105, 19601 (2008)
Zhang, J., Small, M.: Complex network from pseudoperiodic time series: topology versus dynamics. Phys. Rev. Lett. 96, 238701 (2006)
Acknowledgements
B.L. and A.N. acknowledges support from FIS2009-13690 and S2009ESP-1691 (Spain); F.B. from AYA2006-14056, CSD2007-00060, and AYA2010-22111-C03-02 (Spain); A.R. from CONACyT & DGAPA (PAPIIT IN100311)-UNAM (Mexico).
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Communicated by P. Newton.
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Luque, B., Ballesteros, F.J., Núñez, A.M. et al. Quasiperiodic Graphs: Structural Design, Scaling and Entropic Properties. J Nonlinear Sci 23, 335–342 (2013). https://doi.org/10.1007/s00332-012-9153-2
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DOI: https://doi.org/10.1007/s00332-012-9153-2