Quasiperiodic graphs: structural design, scaling and entropic properties

Quasiperiodic graphs: structural design, scaling and entropic properties

B. Luque, F. J. Ballesteros, A. M. Núñez and A. Robledo Dept. Matemática Aplicada y Estadística. ETSI Aeronáuticos, Universidad Politécnica de Madrid, Spain.
Observatori Astronòmic. Universitat de València, Spain.
Instituto de Física y Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Mexico.

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. And finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.

05.45.Ac, 05.90.+m, 05.10.Cc

Quasiperiodicity is observed along time evolution in nonlinear dynamical systems Luque et al. (2011b); Strogatz (1994); Hilborn (1994) and also in the spatial arrangements of crystals with forbidden symmetries SBGC (1984); Schroeder (1991). These two manifestations of quasiperiodicity are rooted in self-similarity and are seen to be related through analogies between incommensurate quantities in time and spatial domains Schroeder (1991). Here we point out that quasiperiodicity can be visualized in a third way: in the graphs generated when the Horizontal Visibility (HV) algorithm Lacasa et al. (2008); Luque et al. (2009) is applied to the stationary trajectories of the universality class of low-dimensional nonlinear iterated maps with a cubic inflexion point, as represented by the circle map Schroeder (1991).

The idea of mapping time series into graphs has been presented in recent works Zhang & Small (2006); Kyriakopoulos & Thurner (2007); Xu et al. (2008); Donner (2010); Donner et al. (2011); Donner (2011); Campanharo et al. (2011) where different approaches have been developed. In particular, the period-doubling bifurcation cascade has been analyzed in the light of the HV formalism Luque et al. (2011, 2011b) and a complete set of graphs, called Feigenbaum graphs, that encode the dynamics of all stationary trajectories of unimodal maps has been provided. The Feigenbaum scenario is one of the three well-known routes to reach chaos in low-dimensional dissipative systems (along with the intermittency route and the quasiperiodicity route) Luque et al. (2011b); Strogatz (1994); Hilborn (1994). In this Letter we characterize the structural, scaling and entropic properties of the graphs obtained when the HV formalism is applied to the quasiperiodic routes to chaos. As we shall see, a Renormalization Group (RG) treatment of such graphs is the instrument that grants access to our main results.

Figure 1: Examples of two standard circle map periodic series with dressed winding number , (top) and (bottom). As can be observed, the order of visits on the circle and the relative values of remain invariant and the associated HV graph is therefore the same in both cases.

We briefly recall that the standard circle map Luque et al. (2011b); Strogatz (1994); Hilborn (1994) is the one-dimensional iterated map given by:


representative of the general class of nonlinear circle maps: , where is a periodic function that fulfills . The HV graphs obtained for this family of maps exhibit universal properties that without loss of generality we explain in the next paragraphs in terms of the standard circle map.

The dynamical variable can be interpreted as a measure of the angle that specifies the trajectory on the unit circle, the control parameter is the so-called bare winding number, and is a measure of the strength of the nonlinearity. The dressed winding number for the map is defined as the limit of the ratio: and represents an averaged increment of per iteration. For trajectories are periodic (locked motion) when the corresponding dressed winding number is a rational number and quasiperiodic when it is irrational. The winding numbers form a devil’s staircase which makes a step at each rational number and remains constant for a range of . For (critical circle map) locked motion covers the entire interval of leaving only a multifractal subset of unlocked.

Figure 2: Six levels of the Farey tree and the periodic motifs of the graphs associated with the corresponding rational fractions taken as dressed winding numbers in the circle map (for space reasons only two of these are shown at the sixth level). (a) In order to show how graph concatenation works, we have highlighted an example using different grey tones on the left side: as , is placed on the left, on the right and their extremes are connected to an additional link closing the motif . (b) Five steps in the Golden ratio route, (thick solid line); (c) Three steps in the Silver ratio route, (thick dashed line).

The resulting hierarchy of mode-locking steps at can be conveniently represented by a Farey tree which orders all the irreducible rational numbers according to their increasing denominators . In the devil’s staircase, , the width of the steps (intervals where is constant) becomes smaller when the denominator increases. Furthermore, if we have two steps with winding numbers and , the largest step between them has a winding number , which is also the irreducible rational number with the smallest denominator. Thus, the Farey tree also orders all mode-locking steps with in the circle map according to their decreasing widths Hao & Zeng (1998).

The HV algorithm assigns each datum of a time series to a node in its associated HV graph, and and are two connected nodes if for all such that . Without loss of generality, we apply the HV algorithm to the superstable orbits of the critical circle map () with an irreducible rational number . Thus, the associated time series always contains as one of its values and has period (cf. Luque et al. (2011b); Strogatz (1994); Hilborn (1994)). The associated HV graph is a periodic repetition of a motif with nodes, of which have connectivity . (Observe that in the map indicates the number of turns in the circle to complete a period). For , the order of visits of positions in the attractors and their relative values remain invariant for a locked region with Hao & Zeng (1998), such that the HV graphs associated with them are the same. In fig. 1 we present an example where the first and the last node in the motif correspond to the largest value in the attractor.

In fig. 2 we depict the associated HV periodic motifs for each in the Farey tree. We observe straightforwardly that the graphs can be constructed by means of the following inflation process: let be a Farey fraction with ‘parents’ , i.e., . The ‘offspring’ graph associated with , can be constructed by the concatenation of the graphs of its parents. By means of this recursive construction we can systematically explore the structure of every graph along a sequence of periodic attractors leading to quasiperiodicity. A standard procedure to study the quasiperiodic route to chaos is fixing and selecting an irrational number . Then, a sequence of rational numbers approaching is taken. This sequence can be obtained through successive truncations of the continued fraction expansion of . The corresponding bare winding numbers provide attractors whose periods grow towards the onset of chaos, where the period of the attractor must be infinite. A well-studied case is the sequence of rational approximations of , the reciprocal of the Golden ratio, that yields winding numbers where is the Fibonacci number generated by the recurrence with and . The first few steps of this route are shown in fig. 2(b): . Within the range one observes trajectories of period and, therefore, this route to chaos consists of an infinite family of periodic orbits with increasing periods of values , . If we denote by the graph associated to in the Golden ratio route, it is easy to prove that the associated connectivity distribution for with and is , , and . In the limit the connectivity distribution at the accumulation point , the quasiperiodic graph at the onset of chaos, takes the form


Figure 3(a) shows that the theoretical degree distribution of the quasiperiodic graph for the route described above is in perfect agreement with the same quantity obtained by applying the HV algorithm to a circle map time series with a dressed winding number . The procedure explained for the Golden ratio can be repeated for the ‘time reversed sequence’: . In this case the ratio converges to in the limit . The connectivity distributions of the graphs are the same as in the Golden ratio route because these graphs are symmetric mirror versions of the former (we use the term ‘time’ because the ‘time reverse’ of a graph from a series generated by a clockwise rotation in the circle map corresponds to the graph from the same but counterclockwise rotation).

The previous results can be interpreted through a suitably-defined Renormalization Group (RG) transformation. We proceed as in previous works Luque et al. (2011, 2011b) and define the RG graph operation as the coarse-graining of every couple of adjacent nodes where one of them has degree into a block node that inherits the links of the previous two nodes. If we continue with the case of the Golden ratio, we first note that and , so the RG flow alternates between the two mirror routes. If we define the operator ‘time reverse’ by , the transformation becomes and . Repeated application of yields two RG flows that converge, for finite, to the trivial fixed point (a graph with ). The accumulation points , the quasiperiodic graphs, act as nontrivial fixed points of the RG flow: and .

The above RG procedure works only in the case of the Golden ratio route. (As a counterexample look at the so-called Silver ratio route shown in fig. 2c). To extend the above formalism to other irrational numbers, we develop the following explicit algebraic version of and apply it to the Farey fractions associated with the graphs,


along with the algebraic analog of the ‘time reverse’ operator . Observe that along the Golden ratio route fractions are always greater than and we can therefore renormalize this route by making


whose fixed-point equation is , with a solution of it.

A straightforward generalization of this scheme is obtained by considering the routes with , , and a natural number. It is easy to see that , which is a solution of the equation . Interestingly, all the positive solutions of the above family of quadratic equations happen to be positive quadratic irrationals in with pure periodic continued fraction representation: ( corresponds to the Golden route). Every fulfills the condition , and, as a result, we have


The transformation can only be applied times before the result turns greater than , so the subsequent application of followed by reversion yields


It is easy to demonstrate by induction that


whose fixed-point equation leads in turn to , with a solution of it. We can proceed in an analogous way for the symmetric case but, as the sense of the inequalities for is reversed, the role of the operators and must be exchanged.

The previous result indicates that graphs must be renormalized via . Again, the iteration of this process yields two RG flows that converge to the trivial fixed point for finite. The quasiperiodic graphs, reached as accumulation points (), act as nontrivial fixed points of the RG flow since .

We observe that for fixed , and from the construction process illustrated in fig. 2(a), it can be deduced that , and . can be obtained from the condition of RG fixed-point invarance of the distribution, as it implies a balance equation whose solution has the form of an exponential tail. The degree distribution for this quasiperiodic graphs is therefore


A perfect agreement between theoretical and numerical results for some examples can be observed in fig. 3(b).

Figure 3: Empty circles stand for theoretical degree distributions of quasiperiodic graphs. Filled values have been obtained by direct application of the HV algorithm to critical circle map series of values. Distributions have been shifted from each other to enhance visualization. From down-up: (a) , . (b) , ; , ; , .

Notably, all the RG flow directions and fixed points described above can be derived directly from the information contained in the degree distribution via optimization of the graph entropy functional . The optimization is for a fixed and takes into account the constrains: , , maximum possible mean connectivity Luque et al. (2011b) and . The degree distributions that maximize can be proven to be exactly the connectivity distributions of equations 2 and 8 for the quasiperiodic graphs at the accumulation points found above. This establishes a functional relation between the fixed points of the RG flow and the extrema of as it was verified for the period-doubling route Luque et al. (2011, 2011b).

We have demonstrated the capability of the HV algorithm for transforming into graph language the universal properties of the route to chaos via quasiperiodicity in low-dimensional nonlinear dynamical systems. The outcome is a novel type of graph architecture where the motifs are the building blocks with which quasiperiodicity is expressed recursively via concatenation. Significantly, the HV formalism leads to analytical expressions for the degree distribution, a function that in all mode-locking regions is essentially exponential. The networks’ scaling properties can be formulated in terms of an ad hoc RG transformation for which the nontrivial graph fixed points capture the features of the quasiperiodic accumulation points. As we have seen, it is through the properties of the RG transformation presented above that the relevant details of the quasiperiodic graphs studied are determined. This class represents all the quasiperiodic attractors reached when irrational winding numbers with pure periodic continued fractions are used as dressed winding numbers. Furthermore, a graph entropy is introduced via the degree distribution and its optimization reproduces the RG fixed points.

By means of the HV algorithm, we have found a connection between pure periodic continued fractions and the degree distribution of their associated quasiperiodic graphs. It seems feasible to generalize our results beyond to periodic continued fractions or any irrational with a pattern in its continued fraction. Finally, it has not escaped our notice that, as we have a one-to-one correspondence between graphs and rational numbers, a possible graph algebra can be explored.

BL and AN acknowledges support from FIS2009-13690 and S2009ESP-1691 (Spain); FB from AYA2006-14056, CSD2007-00060, and AYA2010-22111-C03-02 (Spain); AR from CONACyT & DGAPA (PAPIIT IN100311)-UNAM (Mexico).


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