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

A. M. Núñez

A. Robledo

A.S.L.O. Campanharo

B. Luque

B.-H. Hao

D. Shechtman

F. J. Ballesteros

F. Kyriakopoulos

H.G. Schuster

J. Zhang

L. Lacasa

M. Schroeder

R.C. Hilborn

R.V. Donner

S.H. Strogatz

X. Xu

English

arXiv

A novel class of graphs, here named quasiperiodic, are const ructed 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 Far ey 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 entrop

Luque Serrano, Bartolome

Ballesteros, Fernando J.

Núñez, Ángel M.

Robledo, Alberto

Archivo Digital UPM

Quasiperiodic Graphs: Structural Design, Scaling and Entropic Properties

A novel class of graphs, here named quasiperiodic, are const

ructed 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 Far ey 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 entrop

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

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Quasiperiodic graphs: structural design, scaling and entropic properties
B. Luque1, F. J. Ballesteros2, A. M. Nu´n˜ez1 and A. Robledo3
1 Dept. Matema´tica Aplicada y Estad´ıstica. ETSI Aerona´uticos, Universidad Polite´cnica de Madrid, Spain.
2 Observatori Astrono`mic. Universitat de Vale`ncia, Spain.
3 Instituto de F´ısica y Centro de Ciencias de la Complejidad,
Universidad Nacional Auto´noma de Me´xico, 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.
PACS numbers: 05.45.Ac, 05.90.+m, 05.10.Cc
Quasiperiodicity is observed along time evolution in
nonlinear dynamical systems [1–3] and also in the spa-
tial arrangements of crystals with forbidden symmetries
[4, 5]. 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 [5]. Here we point out that
quasiperiodicity can be visualized in a third way: in the
graphs generated when the Horizontal Visibility (HV) al-
gorithm [6, 7] is applied to the stationary trajectories of
the universality class of low-dimensional nonlinear iter-
ated maps with a cubic inflexion point, as represented by
the circle map [5].
The idea of mapping time series into graphs has been
presented in recent works [8–14] where different ap-
proaches have been developed. In particular, the period-
doubling bifurcation cascade has been analyzed in the
light of the HV formalism [15, 16] and a complete set of
graphs, called Feigenbaum graphs, that encode the dy-
namics 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 inter-
mittency route and the quasiperiodicity route) [1–3]. 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) treat-
ment of such graphs is the instrument that grants access
to our main results.
We briefly recall that the standard circle map [1–3] is
the one-dimensional iterated map given by:
θt+1 = fΩ,K(θt) = θt +Ω− K
2pi
sin(2piθt), mod 1, (1)
representative of the general class of nonlinear circle
maps: θt+1 = fΩ,K(θt) = θt+Ω+K ·g(θt), mod 1, where
g(θ) is a periodic function that fulfills g(θ + 1) = g(θ).
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
FIG. 1: Examples of two standard circle map periodic series
with dressed winding number ω = 5/8, K = 0 (top) and
K = 1 (bottom). As can be observed, the order of visits on
the circle and the relative values of θn remain invariant and
the associated HV graph is therefore the same in both cases.
circle map.
The dynamical variable 0 ≤ θt < 1 can be inter-
preted as a measure of the angle that specifies the tra-
jectory on the unit circle, the control parameter Ω is the
so-called bare winding number, and K is a measure of
the strength of the nonlinearity. The dressed winding
number for the map is defined as the limit of the ra-
tio: ω ≡ limt→∞(θt − θ0)/t and represents an averaged
increment of θt per iteration. For 0 ≤ K ≤ 1 trajecto-
ries are periodic (locked motion) when the correspond-
ing dressed winding number ω(Ω) is a rational number
p/q and quasiperiodic when it is irrational. The wind-
ing numbers ω(Ω) form a devil’s staircase which makes a
step at each rational number ω = p/q and remains con-
stant for a range of Ω. For K = 1 (critical circle map)
locked motion covers the entire interval of Ω leaving only
a multifractal subset of Ω unlocked.
The resulting hierarchy of mode-locking steps atK = 1
can be conveniently represented by a Farey tree which or-
ders all the irreducible rational numbers p/q ∈ [0, 1] ac-
cording to their increasing denominators q. In the devil’s
2FIG. 2: Six levels of the Farey tree and the periodic motifs of the graphs associated with the corresponding rational fractions
p/q taken as dressed winding numbers ω 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 1/3 > 1/4, G(1/3) is placed on the left, G(1/4) on the right and their extremes are connected to an additional link closing
the motif G(2/7). (b) Five steps in the Golden ratio route, b = 1 (thick solid line); (c) Three steps in the Silver ratio route,
b = 2 (thick dashed line).
staircase, ω(Ω), the width of the steps (intervals where
ω is constant) becomes smaller when the denominator q
increases. Furthermore, if we have two steps with wind-
ing numbers p/q and p′/q′, the largest step between them
has a winding number (p+ p′)/(q+ q′), which is also the
irreducible rational number with the smallest denomina-
tor. Thus, the Farey tree also orders all mode-locking
steps with ω = p/q in the circle map according to their
decreasing widths [17].
The HV algorithm assigns each datum θi of a time
series {θi}i=1,2,... to a node i in its associated HV graph,
and i and j are two connected nodes if θi, θj > θn for all n
such that i < n < j. Without loss of generality, we apply
the HV algorithm to the superstable orbits of the critical
circle map (K = 1) with an irreducible rational number
ω(Ω) = p/q. Thus, the associated time series always
contains θ0 = 0 as one of its values and has period q (cf.
[1–3]). The associated HV graph is a periodic repetition
of a motif with q nodes, p of which have connectivity
k = 2. (Observe that p in the map indicates the number
of turns in the circle to complete a period). For K ≤ 1,
the order of visits of positions in the attractors and their
relative values remain invariant for a locked region with
ω = p/q [17], 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 mo-
tifs for each p/q in the Farey tree. We observe straight-
forwardly that the graphs can be constructed by means
of the following inflation process: let p/q be a Farey
fraction with ‘parents’ p′/q′ < p′′/q′′, i.e., p/q = (p′ +
p′′)/(q′ + q′′). The ‘offspring’ graph G(p/q) associated
with ω = p/q, can be constructed by the concatenation
G(p′′/q′′) ⊕ G(p′/q′) of the graphs of its parents. By
means of this recursive construction we can systemat-
ically explore the structure of every graph along a se-
quence of periodic attractors leading to quasiperiodic-
ity. A standard procedure to study the quasiperiodic
route to chaos is fixing K = 1 and selecting an irra-
tional number ω∞ ∈ [0, 1]. Then, a sequence ωn of
rational numbers approaching ω∞ is taken. This se-
quence can be obtained through successive truncations
of the continued fraction expansion of ω∞. The corre-
sponding bare winding numbers Ω(ωn) 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 ω∞ = φ
−1 = (
√
5 − 1)/2 ≃ 0.6180..., the recipro-
cal of the Golden ratio, that yields winding numbers
{ωn = Fn−1/Fn}n=1,2,3... where Fn is the Fibonacci num-
ber generated by the recurrence Fn = Fn−1 + Fn−2 with
F0 = 1 and F1 = 1. The first few steps of this route
are shown in fig. 2(b): ω1 = 1/1, ω2 = 1/2, ω3 =
2/3, ω4 = 3/5, ω5 = 5/8..., ω6 = 8/13... . Within the
range Ω(Fn−1/Fn) one observes trajectories of period Fn
and, therefore, this route to chaos consists of an infinite
family of periodic orbits with increasing periods of val-
ues Fn, n → ∞. If we denote by Gφ−1(n) the graph
associated to ωn = Fn−1/Fn in the Golden ratio route,
it is easy to prove that the associated connectivity dis-
tribution P (k) for Gφ−1(n) with n ≥ 3 and k ≤ n + 1
is Pn(2) = Fn−2/Fn, Pn(3) = Fn−3/Fn, Pn(4) = 0 and
Pn(k) = Fn−k+1/Fn. In the limit n→∞ the connectiv-
ity distribution at the accumulation point Gφ−1(∞), the
quasiperiodic graph at the onset of chaos, takes the form
P∞(k) =


1− φ−1 k = 2
2φ−1 − 1 k = 3
0 k = 4
φ1−k k ≥ 5.
(2)
Figure 3(a) shows that the theoretical degree distribution
3of 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 ω∞ = φ
−1. The pro-
cedure explained for the Golden ratio can be repeated
for the ‘time reversed sequence’: {ωn = Fn−2/Fn =
1−Fn−1/Fn}n=1,2,3,... In this case the ratio converges to
1 − φ−1 in the limit n → ∞. The connectivity distribu-
tions of the graphs {G1−φ−1(n)}n=1,2,3,.. are the same as
in the Golden ratio route because these graphs are sym-
metric 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 corre-
sponds to the graph from the same but counterclockwise
rotation).
The previous results can be interpreted through a
suitably-defined Renormalization Group (RG) transfor-
mation. We proceed as in previous works [15, 16]
and define the RG graph operation R as the coarse-
graining of every couple of adjacent nodes where one of
them has degree k = 2 into a block node that inher-
its the links of the previous two nodes. If we continue
with the case of the Golden ratio, we first note that
R{Gφ−1(n)} = G1−φ−1(n − 1) and R{G1−φ−1(n)} =
Gφ−1(n − 1), so the RG flow alternates between the
two mirror routes. If we define the operator ‘time re-
verse’ by Gφ−1(n) ≡ G1−φ−1(n), the transformation be-
comes R{Gφ−1(n)} = Gφ−1(n−1) and R{G1−φ−1(n)} =
G1−φ−1(n−1). Repeated application of R yields two RG
flows that converge, for n finite, to the trivial fixed point
G0 (a graph with P (2) = 1). The accumulation points
n→∞, the quasiperiodic graphs, act as nontrivial fixed
points of the RG flow: R{Gφ−1(∞)} = Gφ−1(∞) and
R{G1−φ−1(∞)} = G1−φ−1(∞).
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 R and apply it
to the Farey fractions associated with the graphs,
R
(
p
q
)
=
{
R1
(
p
q
)
= pq−p if
p
q <
1
2
R2
(
p
q
)
= 1− q−pp if pq > 12 ,
(3)
along with the algebraic analog of the ‘time reverse’ op-
erator R(x) = 1− R(x). Observe that along the Golden
ratio route fractions are always greater than 1/2 and we
can therefore renormalize this route by making
R
(
Fn−1
Fn
)
= R2
(
Fn−1
Fn
)
=
Fn−2
Fn−1
, (4)
whose fixed-point equation R(x) = x is x2 + x − 1 = 0,
with φ−1 a solution of it.
A straightforward generalization of this scheme
is obtained by considering the routes {ωn =
Fn−1/Fn}n=1,2,3... with Fn = bFn−1 + Fn−2, F0 = 1,
F1 = 1 and b a natural number. It is easy to see that
limn→∞ Fn−1/Fn = (−b +
√
b2 + 4)/2, which is a solu-
tion of the equation x2 + bx − 1 = 0. Interestingly, all
the positive solutions of the above family of quadratic
equations happen to be positive quadratic irrationals in
[0, 1] with pure periodic continued fraction representa-
tion: φ−1b = [b, b, b, ...] = [b¯] (b = 1 corresponds to
the Golden route). Every b > 1 fulfills the condition
Fn−1/Fn < 1/2, and, as a result, we have
R
(
Fn−1
Fn
)
= R1
(
Fn−1
Fn
)
=
Fn−1
(b − 1)Fn + Fn−2 . (5)
The transformation R1 can only be applied (b− 1) times
before the result turns greater than 1/2, so the subse-
quent application of R followed by reversion yields
R(b)
(
Fn−1
Fn
)
= R2
[
R
(b−1)
1
(
Fn−1
Fn
)]
=
Fn−2
Fn−1
. (6)
It is easy to demonstrate by induction that
R
(b−1)
1 (x) =
x
1− (b− 1)x , (7)
whose fixed-point equation R(b)(x) = R2[R
(b−1)
1 (x)] = x
leads in turn to x2+bx−1 = 0, with φ−1b a solution of it.
We can proceed in an analogous way for the symmetric
case ωn = 1−(Fn−1/Fn) but, as the sense of the inequal-
ities for 1/2 is reversed, the role of the operators R1 and
R2 must be exchanged.
The previous result indicates that graphs must be
renormalized via Rb{Gφ−1
b
(n)} = Gφ−1
b
(n − 1). Again,
the iteration of this process yields two RG flows that
converge to the trivial fixed point G0 for n finite. The
quasiperiodic graphs, reached as accumulation points
(n → ∞), act as nontrivial fixed points of the RG flow
since Rb{Gφ−1
b
(∞)} = Gφ−1
b
(∞).
We observe that for fixed b ≥ 2, and from the con-
struction process illustrated in fig. 2(a), it can be de-
duced that P∞(2) = φ
−1
b , P∞(3) = 1 − 2φ−1b and
P∞(k 6= bn + 3) = 0, ∀n ∈ N. P∞(k = bn + 3), n ∈ N
can be obtained from the condition of RG fixed-point in-
varance of the distribution, as it implies a balance equa-
tion P∞(k) = φ
−1
b P∞(k+ b) whose solution has the form
of an exponential tail. The degree distribution P∞(k) for
this quasiperiodic graphs is therefore
P∞(k) =


φ−1b k = 2
1− 2φ−1b k = 3
(1− φ−1b )φ(3−k)/bb k = bn+ 3, n ∈ N
0 otherwise.
(8)
A perfect agreement between theoretical and numerical
results for some examples can be observed in fig. 3(b).
Notably, all the RG flow directions and fixed points
described above can be derived directly from the in-
formation contained in the degree distribution via
optimization of the graph entropy functional H =
−∑∞k=2 P (k) logP (k). The optimization is for a fixed
4k
0 10 20 30 4010
-10
10-8
10-6
10-4
10-2
100
P(k)
FIG. 3: Empty circles stand for theoretical degree distribu-
tions of quasiperiodic graphs. Filled values have been ob-
tained by direct application of the HV algorithm to critical
circle map series of 106 values. Distributions have been shifted
from each other to enhance visualization. From down-up: (a)
ω∞ = [1¯], Ω = 0.606661.... (b) ω∞ = [2¯], Ω = 0.418864...;
ω∞ = [3¯], Ω = 0.323873...; ω∞ = [4¯], Ω = 0.271502....
b and takes into account the constrains: P (2) = φ−1b ,
P (3) = 1 − 2φ−1b , maximum possible mean connectivity〈k〉 = 4 [16] and P (k) = 0 ∀k 6= bn+ 3, n ∈ N. The de-
gree distributions P (k) that maximize H can be proven
to be exactly the connectivity distributions of equations
2 and 8 for the quasiperiodic graphs at the accumula-
tion points found above. This establishes a functional
relation between the fixed points of the RG flow and the
extrema of H as it was verified for the period-doubling
route [15, 16].
We have demonstrated the capability of the HV algo-
rithm for transforming into graph language the univer-
sal 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 quasiperiod-
icity is expressed recursively via concatenation. Signifi-
cantly, 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 ac-
cumulation 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 quasiperi-
odic attractors reached when irrational winding numbers
with pure periodic continued fractions are used as dressed
winding numbers. Furthermore, a graph entropy is in-
troduced via the degree distribution and its optimization
reproduces the RG fixed points.
By means of the HV algorithm, we have found a con-
nection 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 es-
caped our notice that, as we have a one-to-one correspon-
dence 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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