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.
In recent years, a new general framework to make time series analysis has been coined. This framework is based on the mapping of a time series into a network representation and the subsequent graph theoretical analysis of the network, offering the possibility of describing the structure of complex signals and the associated dynamical systems from a new and complementary viewpoint and with a full set of alternative measures. Here, we focus on a specific type of mapping called the horizontal visibility algorithm and, via this approach, we address the specific case of the period-doubling route to chaos. We extend our preliminary results on this topic1 and provide a complete graph theoretical characterization of unimodal iterated maps undergoing period doubling route to chaos that, we show, evidence a universal character. Our approach allows us to visualize, classify, and characterize periodic, chaotic, and onset of chaos dynamics in terms of their associated networks.
The authors acknowledge suggestions from anonymous referees. B.L. and L.L. acknowledge financial support from the MEC and Comunidad de Madrid (Spain) through Project Nos. FIS2009-13690 and S2009ESP-1691. F.J.B. acknowledges support from MEC through Project No. AYA2006-14056, and A.R. acknowledges support from MEC (Spain) and CONACyT and DGAPA-UNAM (Mexican agencies).
II. FEIGENBAUM GRAPHS
A. Mean degree
B. Normalized mean distance
III. PERIOD-DOUBLING ROUTE TO CHAOS: RESULTS
A. Order of visits of stable branches and chaotic bands
B. Topological properties of Feigenbaum graphs along the period-doubling cascade
1. Degree distribution
2. Mean degree and normalized distance
3. Clustering coefficient c(n, k)
4. Higher moments of the degree distribution: Variance
IV. REVERSE BIFURCATION CASCADE OF CHAOTIC BANDS: RESULTS
A. Self-affine properties of chaotic bands: Mean degree and degree distribution
B. Periodic windows: Self-affine copies of the Feingenbaum diagram
V. RENORMALIZATION GROUP APPROACH
A. RG transformation: Definition, flows, and fixed points
B. Crossover phenomenon
A. Entropy optimization and RG fixed points
B. Networkentropy and Pesin theorem
1. Periodic attractors
2. Chaotic attractors
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