^{1}, Lucas Lacasa

^{1,a)}, Fernando J. Ballesteros

^{2}and Alberto Robledo

^{3}

### Abstract

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 topic

^{1}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).

I. INTRODUCTION

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

VI. NETWORKENTROPY

A. Entropy optimization and RG fixed points

B. Networkentropy and Pesin theorem

1. Periodic attractors

2. Chaotic attractors

VII. SUMMARY

### Key Topics

- Bifurcations
- 43.0
- Attractors
- 38.0
- Entropy
- 35.0
- Chaos
- 17.0
- Time series analysis
- 15.0

## Figures

(Color online) Feigenbaum graphs from the Logistic map . The main figure portrays the family of attractors of the Logistic map and indicates a transition from periodic to chaotic behavior at through period-doubling bifurcations. For , the figure shows the merging of chaotic-band attractors where aperiodic behavior appears interrupted by windows that, when entered from their left-hand side, display periodic motion of period with (for , ) that subsequently develops into *m* period-doubling cascades with new accumulation points . Each accumulation point is in turn the limit of a chaotic-band reverse bifurcation cascade with *m* initial chaotic bands, reminiscent of the self-affine structure of the entire diagram. All unimodal maps exhibit a period-doubling route to chaos with universal asymptotic scaling ratios between successive bifurcations that depend only on the order of the nonlinearity of the map, ^{ 19 } the Logistic map belongs to the quadratic case. Adjoining the main figure, we show time series and their associated Feigenbaum graphs according to the HV mapping criterion for several values of where the map evidences both regular and chaotic behaviors (see the text). Inset: Numerical values of the mean normalized distance as a function of mean degree of the Feigenbaum graphs for (associated to the time series of 1500 data after a transient and a step ), in good agreement with the theoretical linear relation (see the text).

(Color online) Feigenbaum graphs from the Logistic map . The main figure portrays the family of attractors of the Logistic map and indicates a transition from periodic to chaotic behavior at through period-doubling bifurcations. For , the figure shows the merging of chaotic-band attractors where aperiodic behavior appears interrupted by windows that, when entered from their left-hand side, display periodic motion of period with (for , ) that subsequently develops into *m* period-doubling cascades with new accumulation points . Each accumulation point is in turn the limit of a chaotic-band reverse bifurcation cascade with *m* initial chaotic bands, reminiscent of the self-affine structure of the entire diagram. All unimodal maps exhibit a period-doubling route to chaos with universal asymptotic scaling ratios between successive bifurcations that depend only on the order of the nonlinearity of the map, ^{ 19 } the Logistic map belongs to the quadratic case. Adjoining the main figure, we show time series and their associated Feigenbaum graphs according to the HV mapping criterion for several values of where the map evidences both regular and chaotic behaviors (see the text). Inset: Numerical values of the mean normalized distance as a function of mean degree of the Feigenbaum graphs for (associated to the time series of 1500 data after a transient and a step ), in good agreement with the theoretical linear relation (see the text).

(Color online) Graphical illustration of the constructive proof of the expression for the mean degree through the consideration of a motif extracted from a periodic series of period *T* = 5. Observe that the second data is the one with the lowest value. By deleting it, the graph loses 2 links. This process is iterated three more times. A total amount of links has been deleted, independently of the inner structure of the *T* = 5 motif.

(Color online) Graphical illustration of the constructive proof of the expression for the mean degree through the consideration of a motif extracted from a periodic series of period *T* = 5. Observe that the second data is the one with the lowest value. By deleting it, the graph loses 2 links. This process is iterated three more times. A total amount of links has been deleted, independently of the inner structure of the *T* = 5 motif.

Periodic Feigenbaum graphs for . The sequence of graphs associated to periodic attractors with increasing period undergoing a period-doubling cascade. The pattern that occurs for increasing values of the period is related to the universal ordering with which an orbit visits the points of the attractor. Observe that the hierarchical self-similarity of these graphs requires that the graph for is a subgraph of that for *n*.

Periodic Feigenbaum graphs for . The sequence of graphs associated to periodic attractors with increasing period undergoing a period-doubling cascade. The pattern that occurs for increasing values of the period is related to the universal ordering with which an orbit visits the points of the attractor. Observe that the hierarchical self-similarity of these graphs requires that the graph for is a subgraph of that for *n*.

(Color online) Graphical illustration that explains how the order of visits to the stable branches of the map induces the structure of the Feigenbaum graphs all along the period-doubling bifurcation cascade ( ).

(Color online) Graphical illustration that explains how the order of visits to the stable branches of the map induces the structure of the Feigenbaum graphs all along the period-doubling bifurcation cascade ( ).

Dots Semi-log plot of the degree distribution of a Feigenbaum graph associated with a time series of data extracted from a Logistic map at the onset of chaos . The straight line corresponds to Eq. (4) , in agreement with the numerical calculation (the deviation for large values of the degree are due to finite size effects).

Dots Semi-log plot of the degree distribution of a Feigenbaum graph associated with a time series of data extracted from a Logistic map at the onset of chaos . The straight line corresponds to Eq. (4) , in agreement with the numerical calculation (the deviation for large values of the degree are due to finite size effects).

(Color online) Aperiodic Feigenbaum graphs for . A sequence of graphs associated with chaotic series after *n* chaotic-band reverse bifurcations, starting at for *n* = 0, when the attractor extends along a single band and the degree distribution does not present any regularity (non-black links). For *n* > 0, the phase space is partitioned in disconnected chaotic bands, and the *n*-th self-affine image of is the *n*-th Misiurewicz point . In all cases, the orbit visits each chaotic band in the same order as in the periodic region . This order of visits induces an ordered structure in the graphs (black links) analogous to that found for the period-doubling cascade.

(Color online) Aperiodic Feigenbaum graphs for . A sequence of graphs associated with chaotic series after *n* chaotic-band reverse bifurcations, starting at for *n* = 0, when the attractor extends along a single band and the degree distribution does not present any regularity (non-black links). For *n* > 0, the phase space is partitioned in disconnected chaotic bands, and the *n*-th self-affine image of is the *n*-th Misiurewicz point . In all cases, the orbit visits each chaotic band in the same order as in the periodic region . This order of visits induces an ordered structure in the graphs (black links) analogous to that found for the period-doubling cascade.

(Color online) Self-affinity in the chaotic region: the two disconnected chaotic bands at are rescaled copies of the first chaotic band . An orbit at makes an alternating journey between both bands.

(Color online) Self-affinity in the chaotic region: the two disconnected chaotic bands at are rescaled copies of the first chaotic band . An orbit at makes an alternating journey between both bands.

(Color online) Zoom of the Feigenbaum diagram close to the period *m* = 3 window. Starting with a period 3 orbit, each one of the stable branches develops into a period-doubling bifurcation cascade with a new accumulation point , beyond which the attractor becomes chaotic, interwoven with periodic windows: each part of the diagram is indeed a rescaled copy of the full Feigenbaum tree. The locations of several Feigenbaum graphs (with the notation defined in the text) within the period three window are depicted.

(Color online) Zoom of the Feigenbaum diagram close to the period *m* = 3 window. Starting with a period 3 orbit, each one of the stable branches develops into a period-doubling bifurcation cascade with a new accumulation point , beyond which the attractor becomes chaotic, interwoven with periodic windows: each part of the diagram is indeed a rescaled copy of the full Feigenbaum tree. The locations of several Feigenbaum graphs (with the notation defined in the text) within the period three window are depicted.

(Color online) Renormalization process and network RG flow structure. (a) Illustration of the renormalization process : a node with degree *k* = 2 is coarse-grained with one of its neighbors (indistinctively) into a block node that inherits the links of both nodes. This process coarse-grains every node with degree *k* = 2 present at each renormalization step. (b) Example of an iterated renormalization process in a sample Feigenbaum graph at a periodic window with initial period *m* = 9 after *n* = 2 period-doubling bifurcations (an orbit of period ). (c) RG flow diagram, where *m* identifies the periodic window that is initiated with period *m* and *ñ* designates the order of the bifurcation, *ñ* = *n* + 1 for period-doubling bifurcations and *ñ* = −(*n* + 1) for reverse bifurcations. denotes the reduced control parameter of the map, and is the location of the accumulation point of the bifurcation cascades within that window. Feigenbaum graphs associated with periodic series ( , *ñ* ) converge to under the RG, whereas those associated with aperiodic ones ( , *ñ* ) converge to . The accumulation point corresponds to the unstable (nontrivial) fixed point of the RG flow, which is nonetheless approached through the critical manifold of graphs at the accumulation points . In summary, the nontrivial fixed point of the RG flow is only reached via the family of the accumulation points, otherwise the flow converges to trivial fixed points for periodic or chaotic regions.

(Color online) Renormalization process and network RG flow structure. (a) Illustration of the renormalization process : a node with degree *k* = 2 is coarse-grained with one of its neighbors (indistinctively) into a block node that inherits the links of both nodes. This process coarse-grains every node with degree *k* = 2 present at each renormalization step. (b) Example of an iterated renormalization process in a sample Feigenbaum graph at a periodic window with initial period *m* = 9 after *n* = 2 period-doubling bifurcations (an orbit of period ). (c) RG flow diagram, where *m* identifies the periodic window that is initiated with period *m* and *ñ* designates the order of the bifurcation, *ñ* = *n* + 1 for period-doubling bifurcations and *ñ* = −(*n* + 1) for reverse bifurcations. denotes the reduced control parameter of the map, and is the location of the accumulation point of the bifurcation cascades within that window. Feigenbaum graphs associated with periodic series ( , *ñ* ) converge to under the RG, whereas those associated with aperiodic ones ( , *ñ* ) converge to . The accumulation point corresponds to the unstable (nontrivial) fixed point of the RG flow, which is nonetheless approached through the critical manifold of graphs at the accumulation points . In summary, the nontrivial fixed point of the RG flow is only reached via the family of the accumulation points, otherwise the flow converges to trivial fixed points for periodic or chaotic regions.

(Color online) Illustrative cartoon incorporating the RG flow of Feigenbaum graphs along the entire Feigenbaum diagram: aperiodic (chaotic or random) series generate graphs whose RG flow converges to the trivial fixed point , whereas periodic series (both in the region and inside periodic windows) generate graphs whose RG flow converges to the trivial fixed point *G*(0,1). The nontrivial fixed point of the RG flow is only reached through the critical manifold of graphs at the accumulation points .

(Color online) Illustrative cartoon incorporating the RG flow of Feigenbaum graphs along the entire Feigenbaum diagram: aperiodic (chaotic or random) series generate graphs whose RG flow converges to the trivial fixed point , whereas periodic series (both in the region and inside periodic windows) generate graphs whose RG flow converges to the trivial fixed point *G*(0,1). The nontrivial fixed point of the RG flow is only reached through the critical manifold of graphs at the accumulation points .

(Color online) Horizontal visibility network entropy *h* and Lyapunov exponent for the Logistic map. We plot the numerical values of *h* and for (the numerical step is , and in each case, the processed time series have a size of data). The inset reproduces the same data but with a rescaled entropy . The surprisingly good match between both quantities is reminiscent of the Pesin identity (see text). Unexpectedly, the Lyapunov exponent within the periodic windows ( inside the chaotic region) is also well captured by *h*.

(Color online) Horizontal visibility network entropy *h* and Lyapunov exponent for the Logistic map. We plot the numerical values of *h* and for (the numerical step is , and in each case, the processed time series have a size of data). The inset reproduces the same data but with a rescaled entropy . The surprisingly good match between both quantities is reminiscent of the Pesin identity (see text). Unexpectedly, the Lyapunov exponent within the periodic windows ( inside the chaotic region) is also well captured by *h*.

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