Estimation of the control parameter from symbolic sequences: unimodal maps with variable critical point.
ABSTRACT The work described in this paper can be interpreted as an application of the order patterns of symbolic dynamics when dealing with unimodal maps. Specifically, it is shown how Gray codes can be used to estimate the probability distribution functions (PDFs) of the order patterns of unimodal maps whose dynamics is controlled by an external parameter. Furthermore, these PDFs depend on the value of the external parameter, which eventually provides a handle to estimate the parameter value from symbolic sequences (in form of Gray codes), even when the critical point depends on the parameter.
- [Show abstract] [Hide abstract]
ABSTRACT: The idea of using chaotic transformations in cryptography is explicit in the foundational papers of Shannon on secrecy systems (e.g., [96]). Although the word “chaos” was not minted till the 1970s [71], Shannon clearly refers to this very concept when he proposes the construction of secure ciphers by means of measure-preserving, mixing maps which depend ‘sensitively’ on their parameters. The implementation of Shannon’s intuitions had to wait till the development of Chaos Theory in the 1980s. Indeed, it was around 1990 when the first chaos-based ciphers were proposed (e.g., [78], [46]). Moreover, in 1990 chaos synchronization [91] entered the scene and shortly thereafter, the first applications to secure communications followed [56, 37]. The idea is remarkably simple: mask the message with a chaotic signal and use synchronization at the receiver to filter out the chaotic signal. The realization though had to overcome the desynchronization induced by the message itself. After this initial stage, the number of proposals which exploited the properties of chaotic maps for cryptographical purposes, grew in a spectacular way.01/2011: pages 257-295; , ISBN: 978-3-642-20541-5 - [Show abstract] [Hide abstract]
ABSTRACT: In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.Communications in Nonlinear Science and Numerical Simulation 07/2014; 19(7):2345–2353. · 2.77 Impact Factor - SourceAvailable from: David Arroyo[Show abstract] [Hide abstract]
ABSTRACT: The idea of closed-loop interaction in in vitro and in vivo electrophysiology has been successfully implemented in the dynamic clamp concept strongly impacting the research of membrane and synaptic properties of neurons. In this paper we show that this concept can be easily generalized to build other kinds of closed-loop protocols beyond (or in addition to) electrical stimulation and recording in neurophysiology and behavioral studies for neuroethology. In particular, we illustrate three different examples of goal-driven real-time closed-loop interactions with drug microinjectors, mechanical devices and video event driven stimulation. Modern activity-dependent stimulation protocols can be used to reveal dynamics (otherwise hidden under traditional stimulation techniques), achieve control of natural and pathological states, induce learning, bridge between disparate levels of analysis and for a further automation of experiments. We argue that closed-loop interaction calls for novel real time analysis, prediction and control tools and a new perspective for designing stimulus-response experiments, which can have a large impact in neuroscience research.PLoS ONE 01/2012; 7(7):e40887. · 3.73 Impact Factor
Page 1
arXiv:0812.2331v1 [nlin.CD] 12 Dec 2008
Estimation of the control parameter from symbolic sequences: Unimodal maps with
variable critical point
David Arroyo,1, ∗Gonzalo Alvarez,1and Jos´ e Mar´ ıa Amig´ o2
1Instituto de F´ ısica Aplicada, Consejo Superior de Investigaciones Cient´ ıficas, Serrano 144—28006 Madrid, Spain
2Centro de Investigaci´ on Operativa, Universidad Miguel Hern´ andez,
Avda.de la Universidad s/n, 03202 Elche, Spain
The work described in this paper can be interpreted as an application of the order patterns of
symbolic dynamics when dealing with unimodal maps. Specifically, it is shown how Gray codes can
be used to estimate the probability distribution functions (PDFs) of the order patterns of parametric
unimodal maps. Furthermore, these PDFs depend on the value of the parameter, what eventually
provides a handle to estimate the parameter value from symbolic sequences (in form of Gray codes),
even when the critical point depends on the parameter.
In this paper, the order patterns of unimodal
maps are studied. It is shown how to construct
order patterns of unimodal maps from their sym-
bolic dynamics with respect to the partition of
the state space introduced by the critical point.
Finally, it is shown that for a subclass of para-
metric unimodal maps, the study of those order
patterns allows to estimate the parameter of the
map that has generated the symbolic sequence.
I.INTRODUCTION
Sarkovskii’s theorem shows that order and dynamics
are intertwined in one-dimensional intervals. It is there-
fore not surprising that the study of the ordinal structure
of deterministic time series gives valuable information on
the underlying dynamical system. This work focuses on
the reconstruction of the so-called order patterns of cer-
tain unimodal maps, from “coarse-grained”orbits in form
of 0-1 sequences: 0 if the corresponding iterate lies to the
left of the critical point, and 1 otherwise. Such binary
sequences will be called Gray codes. The relationship be-
tween the Gray codes of parametric unimodal maps and
the value of the parameter that controls a particular dy-
namic, was shown in [1, 2, 3]. Other important tool for
the understanding of one-dimensional dynamical systems
is the study of their order patterns [4]. Indeed, order pat-
terns allow to distinguish chaos from white noise, and can
provide useful information on the parameter or param-
eters controlling the dynamic of chaotic systems. The
main goal of this paper is to estimate the control param-
eter of unimodal maps by means of their order patterns
alone, even when the exact values of their orbits are not
accessible but only the corresponding Gray codes.
The rest of the paper is organized as follows. First of
all, the general framework is set in Sect. II. In Sect. III,
the concept of order pattern is introduced, and its depen-
dence on the control parameter is analyzed for the logistic
∗Electronic address: david.arroyo@iec.csic.es
and the skew tent maps. Sect. IV summarizes the theory
on Gray codes. How the order patterns of unimodal maps
are obtained using Gray codes is explained in Sect. V; its
application to control parameter estimation is explained
in Sect. VI. The results presented in this paper are re-
capitulated in Sect. VII, where some final comments are
also included.
II.SCENARIO
The work described in this paper focuses on a class of
unimodal maps, hereafter denoted as F. A map f : I →
I, where I = [a,b] ⊂ R, a < b, belongs to the class F if
it satisfies the following conditions.
1. f is continuous.
2. f(a) = f(b) = a.
3. f reaches its maximum value fmax≤ b in the subin-
terval [am,bm] ⊂ I, am≤ bm.
4. f(fmax) < xc, where xcis the middle point of the
interval [am,bm], i.e., xc=am+bm
2
.
5. f(xc) > xc.
6. f is strictly increasing function on [a,am] and
strictly decreasing on [bm,b].
The class F includes maps defined in a parametric way,
say, fλ(x) = ϕ(λ,x), where x ∈ I = [a,b], λ ∈ J ⊂ R is
called the control parameter, and ϕ is a self-map of I×J.
Two different situations are considered in this paper:
1. The control parameter determines the maximum
value of the map. In this case, the parametric func-
tion fλis given by
fλ(x) = λF(x),(1)
where F ∈ F and F(xc) = Fmax. The subclass of
maps fλ∈ F complying with this description will
be denoted by F1.
Page 2
2
2. The control parameter is the value of the critical
point, i.e., xc= λ. This leads to a new subclass of
maps F2.
00.2 0.40.60.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
f(k)
λ(x)
f(1)(x)
f(0)(x)
f(3)(x)
f(2)(x)
[0,1,2,3]
?
[0,1,3,2]
[0,3,1,2]
??
[3,0,1,2]
?
[0,3,1,2]
?
[0,2,1,3]
?
[2,0,3,1]
?
[2,3,0,1]
?
[2,0,3,1]
?
[2,0,1,3]
?
[3,1,0,2]
?
[1,3,2,0]
[1,2,3,0]
[1,2,0,3]
???
[1,2,3,0]
?
FIG. 1: f(k)
patterns of length 4 for the logistic map when λ = 4.
λ(x) for k = 0,1,2,3 and the corresponding order
III.ORDER PATTERNS
Given a closed interval I ⊂ R and a map f : I → I
, the orbit of (the initial condition) x ∈ I is defined as
the set Of(x) = {fn(x) : n ∈ N0}, where N0= {0}∪N =
{0,1,...}, f0(x) = x and fn(x) = f?fn−1(x)?. Orbits
of length L), which are permutations of the elements
{0,1,...,L − 1}, L ≥ 2. We write π = [π0,π1,...,πL−1]
for the permutation 0 ?→ π0,...,L − 1 ?→ πL−1.
Definition 1 (Order pattern). The point x ∈ I is said
to define (or realize) the order L-pattern π = π(x) =
[π0,π1,...,πL−1] if
are used to define order L-patterns (or order patterns
fπ0(x) < fπ1(x) < ... < fπL−1(x).(2)
Alternatively, x is said to be of type π. The set of all
possible order patterns of length L is denoted by SL.
For further reference, it is convenient to assign an inte-
ger number to each order pattern. This can be made, for
instance, by means of the Trotter-Johnson algorithm [5].
The order patterns of length 4 along with their “ordering
numbers”, are shown in Table I.
As emphasized in [6], there always exist order L-
patterns with sufficiently large L that are not realized
in any orbit of f ∈ F. These order patterns are called
forbidden patterns, whereas the rest of order patterns are
called allowed patterns. In general, if fλ is a family of
self-maps of the closed interval I ⊂ R parameterized by
λ ∈ J ⊂ R (as it occurs for fλ∈ F1,F2), and the set Pπ
is defined as
Pπ= {x ∈ I : x is of type π},
where π ∈ SL, then Pπdepends on fλand, consequently,
on λ. According to the ergodic theorem [7, p. 34], if fλis
ergodic with respect to the invariant measure µ, then the
orbit of x ∈ I visits the set Pπ with relative frequency
µ(Pπ), for almost all x with respect to µ. As a result, it
is possible to study the dependence of Pπon λ by count-
ing and normalizing the occurrences of π in sliding win-
dows of width L along Ofλ(x), x being a ‘typical’ initial
condition. In the following two subsections this is done
experimentally with the logistic map (as representative
of F1) and with the skew tent map (as representative of
F2). Since we are primarily interested in the relation be-
tween the probabilities µ(Pπ) (or relative frequencies) of
order patterns π ∈ SLand the control parameter λ of the
map considered, we will refer to it as the λ-distribution
function (in short: λ-DF) of π, since they are related to
the probability distribution functions (we fix π instead of
fixing λ).
(3)
FIG. 2: Relative frequency of the order patterns realized by
the logistic map when L = 4 and λ ∈ [3.7,4].
A.Order patterns for the logistic map
The logistic map, defined as
fλ(x) = λx(1 − x),(4)
for x ∈ [0,1] and λ ∈ [1,4], belongs to F1. The logistic
map with λ = 4 was studied in [6, 8] from the ordinal
point of view. In Fig. 1 the allowed order 4-patterns
Page 3
3
# Order pattern # Order pattern # Order pattern # Order pattern
0[0, 1, 2, 3]1 [0, 1, 3, 2]
4[3, 0, 2, 1]5[0, 3, 2, 1]
8 [2, 0, 1, 3]9 [2, 0, 3, 1]
12[3, 2, 1, 0] 13 [2, 3, 1, 0]
16 [1, 2, 0, 3] 17[1, 2, 3, 0]
20 [3, 1, 0, 2]21[1, 3, 0, 2]
2
6
10
14
18
22
[0, 3, 1, 2]
[0, 2, 3, 1]
[2, 3, 0, 1]
[2, 1, 3, 0]
[1, 3, 2, 0]
[1, 0, 3, 2]
3
7
11
15
19
23
[3, 0, 1, 2]
[0, 2, 1, 3]
[3, 2, 0, 1]
[2, 1, 0, 3]
[3, 1, 2, 0]
[1, 0, 2, 3]
TABLE I: Order patterns of length four.
for the logistic map with λ = 4 are shown. For this
value of the control parameter there exist twelve allowed
order patterns. However, the main goal of this paper is to
analyze the relationship between the control parameter of
maps in F1or F2, and their order patterns, what calls for
the distributions of allowed patterns for different values of
λ. Figure 2 depicts the relative frequencies of each order
4-pattern for λ ∈ [3.7,4], the patterns being labeled as
in Table I. To be more specific, for every λ, a sufficiently
long orbit was generated, the occurrences of the different
order patterns were counted using a sliding window of
width 4, and finally the counts obtained were normalized
by the number of windows. These results are estimates of
the probabilities for the corresponding order patterns to
occur. Let us point out that, since the physical invariant
measure of the logistic map is only known for λ = 4,
numerical estimation of those probabilities is the most
we can hope for. More importantly for us, we conclude
from Fig. 2 that it is very difficult to infer the value of
λ ∈ [3.7,4] from the λ-DF of order patterns of length 4.
B. Order patterns for the skew tent map
The skew tent map, given by
fλ(x) =
?
x/λ,
(1 − x)/(1 − λ), if λ ≤ x ≤ 1,
if 0 ≤ x < λ,
(5)
for x ∈ [0,1] and λ ∈ (0,1), belongs to the subclass F2,
comprised of those maps of F parameterized by the crit-
ical point. Furthermore, for the skew tent map fλ, the
maximum value fλ(xc) = fλ(λ) = 1 is independent from
λ (see Fig. 3). Contrarily to the logistic map, the skew
tent map does possess a known ergodic invariant measure
for all λ ∈ (0,1), namely, the Lebesgue measure on [0,1].
Hence, if Pπis given by Eq. (3) with I = [0,1], the rela-
tive frequency of the order pattern π in a typical orbit of
the skew tent map, coincides with the Lebesgue measure
of Pπ, which can be determined analytically. The easiest
case corresponds to the order pattern π = [0,1,...,L−1],
since then Pπis an open interval whose left endpoint is 0
and whose right endpoint is the leftmost intersection be-
tween f(L−1)
λλ
. The relative frequencies of the
order patterns of length 4, numbered according to Table
I, are depicted in Fig. 4. In particular, the length of the
and f(L−2)
interval P[0,1,2,3]=: (0,φ4(λ)) is determined by the first
intersection between f(2)
λ(x) and f(3)
λ(x):
φ4(λ) =
λ2
2 − λ.
(6)
Therefore, the λ-DF of π = [0,1,2,3] (pattern #0) is
given by φ4(λ); see Fig. 5(a) for the graphical represen-
tation of φ4(λ). The fact that the function φ4(λ) is bijec-
tive entails the possibility of estimating λ via the relative
frequency of the order pattern [0,1,2,3].
Up to this point it has been assumed that the orbits
of the various maps considered, were accessible. From a
more practical point of view, it is also relevant to know
whether order patterns can be still determined using less
information about the orbits. This is the case, for in-
stance, when dealing with the symbolic dynamic asso-
ciated to a generating partition of the state space. In
particular, the orbits of maps of F can be transformed
into binary sequences by the procedure described in [1].
In the next section it is explained how to build order
patterns from those binary sequences.
IV. GRAY CODES AND UNIMODAL MAPS
Symbolic dynamics has been thoroughly studied in the
context of unimodal maps since the seminal contribution
of Metropolis et al. in [1]. In [3] Gray codes were used as
a more intuitive way of understanding and applying the
ideas of [1]. The connection between both approaches can
be mathematically established with the aid of results in
[1, 2, 9], as pointed out in [10]. In this section, we address
the ordinal structure of Gray codes.
For a unimodal map f defined on the interval I =
[a,b], any finite orbit {fn(x) : 0 ≤ n ≤ N − 1}
can be transformed into a binary sequence GN(f,x) =
g(f0(x)) g(f1(x)) ...g(fN−1(x)), where g is the step
function
g(x) =
?
0
1
if x < xc,
if x ≥ xc.
(7)
As x increases from the left endpoint a to the right end-
point b, the interval I can be partitioned into 2Nsubin-
tervals I(N)
j
, 1 ≤ j ≤ 2N, each subinterval containing
Page 4
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1
f(k)
λ(x)
x
(a)λ = 0.3
00.20.40.60.81
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0.5
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0.7
0.8
0.9
1
f(k)
λ(x)
x
(b)λ = 0.7
FIG. 3: The first four iterations of f(x) and the corresponding order patterns of length 4 for the skew tent map, i.e., f(k)
for k = 0,1,2,3.
λ(x)
FIG. 4: Relative frequencies of the order patterns of length
L = 4 realized by the skew tent map.
those x ∈ I whose orbits have resulted into a given bi-
nary sequence GN(f,x). That is, (i) I(N)
j ?= k, (ii) I = I(N)
sequences GN(f,x) obtained for each x ∈ I(N)
same. Moreover, the sequences GN(f,x1) for x1∈ I(N)
and GN(f,x2) for x2∈ I(N)
in one bit. Therefore, if we label the 2Nsubintervals
j
∩ I(N)
j
= ∅ for
1
∪I(N)
2
···∪ I(N)
2N , and (iii) the binary
j
are the
j
j+1, 1 ≤ j ≤ 2N− 1, differ only
00.5
λ
(a)
1
0
0.2
0.4
0.6
0.8
1
Order pattern frequency
00.20.40.60.8
0.01
0.02
0.03
0.04
λ
(b)
Order pattern frequency
00.5
λ
(c)
1
0
0.02
0.04
0.06
0.08
0.1
Order pattern frequency
00.5
λ
(d)
1
0
0.02
0.04
0.06
0.08
Order pattern frequency
FIG. 5: Order pattern frequency for the skew tent map and
L = 4 (a) order pattern #0; (b) order pattern #1; (c) order
pattern #2; (d) order pattern #3.
I(N)
j
continuous subintervals will have only one bit flipped.
For the sake of illustration, let us consider the skew
tent map with λ = 0.5. In Fig. 6, the division of I =
[0,1] into the subintervals I(N)
corresponding binary sequence of length N, is shown for
with the 2Nsequences GN(f,x), then the labels of
j
, each labeled with the
Page 5
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Rank Binary code Gray code
0
1
2
3
4
5
6
7
000
001
010
011
100
101
110
111
000
001
011
010
110
111
101
100
TABLE II: Correspondence between Gray codes and binary
codes for three bits.
N = 1,2,3. The separation points of the subintervals
I(N)
j
are the solutions of the equations
fn−1
1/2(x) =1
2, 1 ≤ n ≤ N.(8)
If, furthermore, GN is the set of all binary sequences
of length N produced by a map f ∈ F, then it is pos-
sible to endow GN with a linear order as follows. Given
GN(f,x1) ?= GN(f,x2), let i be the first index such that
g(fi(x1)) ?= g(fi(x2)). Depending on the value of i, we
distinguish three cases:
- If i = 0 then GN(f,x1) < GN(f,x2) if and only if
g(x1) < g(x2).
- If i > 0 and Gi(f,x1) = Gi(f,x2) contains an even
number of 1’s, then GN(f,x1) < GN(f,x2) if and
only if g(fi(x1)) < g(fi(x2)).
- If i > 0 and Gi(f,x1) = Gi(f,x2) contains an odd
number of 1’s, then GN(f,x1) < GN(f,x2) if and
only if g(fi(x1)) > g(fi(x2)).
Gray codes are well known in the context of commu-
nication theory. The Gray codes of length 3 are shown
in Table II. The main characteristic of the Gray codes is
that two consecutive codes differ in only one bit. More-
over, the order of Gray codes is equivalent to the order
in GN(check Table II for N = 3). As a consequence, any
binary sequence GN(f,x) can be interpreted as a Gray
code of length N [3], and will be called a Gray code here-
after. Finally, the order of the Gray codes derived from
any unimodal map belonging to F is directly linked to
the order in R of the points x ∈ I. Indeed, it is proven
in [9, Lemma 4.1] that GN(f,x1) < GN(f,x2) for some
N ≥ 1, implies x1< x2. This is illustrated in Fig. 6.
V. GRAY CODES AND ORDER PATTERNS
FOR UNIMODAL MAPS
In this section the analysis focuses on the parametric
unimodal maps of the subclasses F1or F2. In section III
we elaborated on the dependence of the order patterns
f(1)(x)
f(2)(x)
x x
xc
a
•
b
•
xc
•
i=0
I(1)
1
I(1)
2
•
i=1
I(2)
1
•
I(2)
2
•
I(2)
3
•
I(2)
4
•
•
i=2
I(3)
1
•
I(3)
2
•
I(3)
3
•
I(3)
4
•
I(3)
5
•
I(3)
6
•
I(3)
7
•
I(3)
8
•
000
0 0
001
0
011
0 1
010110
1 1
111
1
101
1 0
100
1
FIG. 6: Symbolic intervals for different iterations of the skew
tent map for λ = 0.5.
3.73.753.8 3.85
λ
3.93.954
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pattern assigment error rate
FIG. 7: Error rate for the pattern assignment based on Gray
codes with respect to the one based on the orbit of the logistic
map. The length of order patterns is L = 4, the length of the
considered Gray codes is N = 100 and the number of samples
is 10000.The perfect estimation of the PDF of the order
patterns of the logistic map is possible for those values of λ
leading to aperiodic binary sequences or to binary sequences
with period larger than 4, i.e., the length of the considered
order patterns.
allowed for those maps with respect to the control pa-
Page 6
6
rameter. Specifically, we estimated the probabilities of
order 4-patterns by their relative frequencies in orbits of
the logistic map (Fig. 2) and of the skewed tent map (see
Fig. 4) with different parameter settings. Our next goal
is to reproduce the same dependencies not from the exact
values of the orbit point (“sharp orbit”), but from the bi-
nary sequence built as explained in the previous section
(“coarse-grained orbit”). As discussed in that section,
the definition domain I of f ∈ F splits in 2Nsubinter-
vals when Gray codes of length N are considered. We
show next that the order patterns of f can also be ob-
tained comparing Gray codes obtained from its orbits.
Let GM(f,x) = g0g1...gM−1, gi∈ {0,1}, be the Gray
code of length M of x ∈ I.
together with the points x ∈ I, are linearly ordered
and, moreover, their order relations are equivalent (i.e.,
G∞(f,x1) < G∞(f,x2) iff x1 < x2), we can expect to
obtain useful information about the order patterns real-
ized by the sharp orbit Of(x) from the order patterns
realized by the coarse-grained orbit GM(f,x), M ≥ 2.
The procedure is as follows.
Since the Gray codes,
1. Divide the Gray code of length M, GM(f,x), into
M − N + 1 Gray codes of length N < M us-
ing a sliding window of length N.
first Gray code derived from GM(f,x) is G0=
g0g1...gN−1 = GN(f,x), the second Gray code
is G1
= g1g2...gN
= GN(f,f(x)), ..., and
the (M − N + 1)-th Gray code is GM−N
gM−NgM−N+1...gM−1= GN(f,fM−N(x)).
Thus, the
=
2. For i = 0,1,...,M − N − L + 1, build groups of
L consecutive Gray codes GiGi+1...Gi+L−1. The
i-th group defines then the order L-pattern π =
π(i) = [π0,π1,...,πL−1] if
Gi+π0< Gi+π1< ... < Gi+πL−1.
The order patterns derived using Gray codes need not
have, in general, similar λ-DFs to those derived from
the sharp orbits. Indeed, order patterns defined by Gray
codes of length N are built upon the comparison of subin-
tervals I(N)
j
⊂ I (see Sect. IV), rather than comparing
points of I. The width of the intervals I(N)
as the length N of the sliding window increases in such
a way that when N → ∞, each one of those intervals
converges to a single real number. As a result, the error
in the calculation of the order patterns from Gray codes
is expected to reduce as N increases. In the context of
finite-precision computation, the minimum value of N
necessary to get a reliable approximation of the λ-DF of
an order pattern is related to the precision of the arith-
metic used. Again, this quantization error decreases as
N increases and, consequently, a large value of N may be
necessary to assure a good approximation of the λ-DF.
Another source of divergences between λ-DFs and their
numerical estimation via finite-length Gray codes maybe
non-ergodicity or even poor ergodicity. As a matter of
j
decreases
fact, remember that the estimation of the probability
µ(Pπ) by the relative frequency of π ∈ SLin finite orbits
of a µ-preserving map, hinges on the ergodic theorem.
If, furthermore, the convergence of relative frequencies
to probabilities in the orbits of an ergodic map with re-
spect to µ, is very slow, a good estimation would require
exceedingly long sequences —this is what we mean by
“poor ergodicity”. These errors are shown in Figs. 7 and
8 for the logistic and the skew tent maps, respectively,
with π = [0,1,2,3], M = 10104, and N = 100. In the
first case, the value of λ lies within the period-3 window
of the logistic map. In the second case, poor ergodic-
ity is expected for values of λ close to 0 and 1. The
asymmetry in the error distribution is due to the fact
that for λ ≃ 1, the tent map looks like the identity in
most of I = [0,1], hence P[0,1,2,3]covers most of I. This
makes [0,1,2,3] to be the most frequent order 4-pattern
even when its frequency is calculated using Gray codes.
Comparison of Figs. 5 and
of the Gray code-based method for the first four order
4-patterns (see Table II) of the skew tent map .
9 illustrates the accuracy
00.20.40.60.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
Pattern assigment error rate
FIG. 8: Error rate for the pattern assignment based on Gray
codes with respect to the one based on the orbit of the skew
tent map. The length of order patterns is L = 4, the length
of the considered Gray codes is N = 100 and the number
of samples is 10000. A value of the control parameter above
0.2 guarantees a perfect estimation of the PDF of the order
patterns of the skew tent map.
VI. ESTIMATION OF THE CONTROL
PARAMETER FOR UNIMODAL MAPS WITH
CRITICAL POINT DEPENDING ON THE
CONTROL PARAMETER
The main characteristic of maps in F2 is that the
control parameter λ determines the value of the critical
point. Furthermore, from our discussion above, we ex-
pect that the relation between the control parameter and
the allowed order patterns of the corresponding dynam-
ics is specially simple for the pattern π = [0,1,...,L−1].
Page 7
7
0 0.5
λ
1
0
0.2
0.4
0.6
0.8
1
Order pattern frequency
00.5
λ
1
0
0.01
0.02
0.03
0.04
Order pattern frequency
00.5
λ
1
0
0.02
0.04
0.06
0.08
0.1
Order pattern frequency
00.5
λ
1
0
0.02
0.04
0.06
0.08
0.1
Order pattern frequency
FIG. 9: Relative frequency of order patterns of the skew tent
map using Gray codes, when L = 4, N = 100 and the se-
quences are 10104-bit long: (a) order pattern #0; (b) order
pattern #1; (c) order pattern #2; (d) order pattern #3.
Clearly, if the λ-DF of this pattern is 1-to-1, then λ can
be pinpointed from that distribution function; otherwise,
the possible values of λ can be reduced to a few candi-
dates, what can be also acceptable in applications like
cryptanalysis. In turn, λ-DFs can be approximated via
Gray codes, without previous knowledge of the critical
point of the map.The bottom line is that the con-
trol parameter of a map in F2 can be estimated from
their coarse-grained orbits (in form of Gray codes). The
specifics depend on the map.
As an example, consider the skew tent map again. For
this map, the interval P[0,1,...,L−1], i.e., the set of points
x ∈ [0,1] of type [0,1,...,L−1], is determined by the left-
most intersection of the iterates fL−2
λ
and fL−1
λ
, where
fn
λ(x) =
?
x/λn,
(λn−1− x)/λn−1(1 − λ),
if 0 ≤ x ≤ λn,
if λn≤ x ≤ λn−1.
(9)
Hence P[0,1,...,L−1]= [0,φL(λ)], with
φL(λ) =λL−2
2 − λ.
(10)
Since this function is 1-to-1 in the interval 0 ≤ λ ≤ 1 for
L ≥ 2, with φ2(0) = 1/2, φL≥3(0) = 0, and φL≥2(1) = 1,
it allows to estimate λ by estimating φL(λ) —the length
of P[0,1,...,L−1]. Now, from the equation
d
dλφL(λ) =
λL−3
(2 − λ)2[2(L − 2) − (L − 3)λ] =
?
L − 1,
=
0,if λ = 0,
if λ = 1,
(11)
it follows that φL(λ) is a ∪-convex function on 0 ≤ λ ≤ 1
for L ≥ 2, that converges to 0 on 0 ≤ λ < 1 as L → ∞.
Therefore, the higher L the worse φL(λ) discriminates
different values of λ. Consequently, L = 3,4 are the best
choices for a quality estimation of λ.
On the other hand, the ergodicity of the skew tent map
permits to estimate the length of P[0,1,...,L−1]by estimat-
ing the relative frequency of the π = [0,1,...,L−1] in a
typical sharp orbit of the map —or, as we intent, in a typ-
ical coarse-grained orbit. In the latter case, the choice for
the parameter N, the width of the sliding window down
the Gray codes (Sect. V), must be also analyzed. The
minimum value of N to get a good reconstruction of the
λ-DF of the order patterns, Nmin, depends on the pre-
cision of the arithmetic used, but it also depends on the
Lyapunov exponent of the map. If floating point double-
precision arithmetic is implemented, then Nmin can be
determined as function of λ by comparing pairs of sym-
bolic sequences generated from the same initial condition
and control parameters λ1 and λ2 such that |λ2− λ1|
equals the spacing of floating point numbers. As it is
shown in Fig. 10, the value of Nmin increases with the
Lyapunov exponent for the skewed tent map.
Summing up, the estimation of the control parameter
λ ∈ (0,1) of the skew tent map fλ, Eq. (5), can be done
by counting and normalizing the occurrences of the order
pattern [0,1,...,L − 1], ideally for L = 3 or 4, in a sta-
tistically significant sample of orbit segments of fλ. This
follows from the following properties: (i) fλis ergodic for
all λ, and (ii) the fλ-invariant measure of P[0,1,...,L−1](in
this case, the length of the interval P[0,1,...,L−1]) depends
bijectively on λ. In a practical context though, finite
precision machines are used, and this entails, in general,
numerical degradation, this meaning that the computed
orbits, whether of chaotic or non-chaotic maps, depart
from the real ones. In the case of a very long orbit of
a chaotic map, the deviation of the numerical simulation
(locally measured by the Lyapunov exponent of the map)
will be severe; in such cases, it is preferable to have many
shorter orbits instead. Even worse, all orbits computed
with finite precision are eventually periodic. This distor-
tion of the dynamics, due to finite numerical precision
and dependence on initial conditions, implies the general
impossibility of obtaining orbits and invariant measures
in an exact way. As a matter of fact, all this carries over
to symbolic dynamics.
To verify this issue in the case of coarse-grained orbits,
a sample of Gray codes of the skew tent map, each one
with the same length but with a different initial condi-
tion, was generated for every value of λ. The underlying
Page 8
8
00.20.40.60.81
10
1
10
2
10
3
10
4
λ
Nmin
(a)
0.10.2 0.30.40.5
λ
0.60.70.80.9
0.1
0.2
0.3
0.4
0.5
0.6
Lyapunov exponent
(b)
FIG. 10: Dependence of the width of the sliding window with
respect to the rate of divergence:(a) Minimum width of the
sliding window necessary for the reconstruction of the PDF
of the order patterns from the symbolic sequences of the skew
tent map; (b) Lyapunov exponent of the skew tent map.
sharp orbits were computed with double precision float-
ing point arithmetic. From this sample of Gray codes,
the corresponding λ-DFs of the order patterns of length
L = 4 were obtained. The λ-DF of the order pattern
[0,1,2,3] (#0 for short) was calculated as the mean value
of the λ-DFs obtained from the various initial condi-
tions. This average value is compared to the exact λ-DF,
φ4(λ) = λ2/(2−λ), in Fig. 11, along with the correspond-
ing standard deviation.
Fig. 11 spells out that, in the context of finite precision
computation, the perfect recovery of the control param-
eter value using the λ-DF of order pattern #0, is not
feasible in general if one can only resort to Gray codes.
However, it is possible to locate λ up to an uncertainty
interval. The width of this interval can be upper bounded
by the standard deviation of the λ-DF of the order pat-
tern #0 since, according to Fig. 11, it is bigger than the
average error in the estimation of φ4(λ) for every value
of λ . Therefore, the estimation of the control parameter
0.2 0.3 0.40.50.6 0.70.80.9
0.005
0.01
0.015
0.02
0.025
λ
Standard deviation of the PDF of order pattern #0
Average deviation from λ2/(2−λ)
FIG. 11: Average deviation and standard deviation in the
estimation of the PDF of the order pattern #0 for the skew
tent map with respect to λ2/(2 − λ).
comprises two stages:
1. An estimation of λ is performed by dividing the
given Gray code, {gi}M−1
large enough set of disjoint subsequences of length
N ≫ 4, say, {gk·N+i}N−1
⌊M − N⌋ − 1. For each such binary subsequence,
a value of φ4(λ) is then computed as the relative
frequency of the order pattern [0,1,2,3] using, of
course, the Gray ordering (Sect. IV).
the mean value of the resulting φ4(λ)’s.
φ4(λ) = λ2/(2 − λ), Eq. (6), it follows that the
control parameter is estimated as
4(¯ x) =−¯ x +√¯ x2+ 8¯ x
i=0, gi ∈ {0,1}, into a
i=0
for k = 0,1,...,K =
Let ¯ x be
From
ˆλ = φ−1
2
.(12)
2. If σ is the standard deviation of the φ4(λ) sampling,
then
λ ∈ (φ−1
4(¯ x − σ),φ−1
4(¯ x + σ)).(13)
The specifics of this procedure refers to the skew tent
map, but the general strategy is the same, once a bijective
λ-DF of an order pattern is exactly known.
In order to establish the accuracy of the procedure,
some numerical simulations with the skew tent map were
done. For every value of the control parameter, a group
of 100 different initial conditions were used in the gener-
ation of the corresponding Gray codes. For each of these
binary sequences, the control parameter λ was estimated
as just explained. The mean error of the estimation is
shown in Fig. 12. The average error lies always above
10−4, and can only be reduced by the implementation
of the procedure with extended-precision arithmetic li-
braries. To prove this claim, the case of the symmetric
Page 9
9
0.1 0.20.30.40.50.6 0.70.80.91
10
−4
10
−3
10
−2
λ
Mean error value (Logarithmic scale)
FIG. 12: Mean error value in the estimation of the control
parameter of the skew tent map.
tent map, i.e., the skew tent map for λ = 1/2, will be
now considered. For the symmetric tent map the arith-
metic is exact. Indeed, if 0.x0x1...xM, xi∈ {0,1}, is the
expansion to base 2 of x ∈ [0,1], i.e.,
x =
M
?
n=0
xn
2n+1,(14)
(numbers with finite binary expansions are called dyadic
rationals), then the action of the symmetric tent map
amounts to a zero-bit dependent left shift, to wit:
f1/2(0.x0x1...xn...xM) =
?
0.x∗
=
0.x1x2...xn+1...xM−1xM,
1x∗
if x0= 0,
if x0= 1,(15)
2...x∗
n+1...x∗
M−1xM,
where x∗
with M bits and xM = 1, the orbit of x collapses to 0
after M iterations of f1/2, so M can be considered the
effective length of the orbits to be used in an estimation
of λ = 1/2. For L = 3, the relative frequency of the or-
der pattern #0 ([0,1,2] in this case) was determined for
a large set of random initial conditions x and increasing
orbit lengths M. The convergence in average of this rel-
ative frequency to φ3(1/2) = 1/3 (see Eq. (10)) as M
increases, is confirmed by Fig. 13. At the same time, the
variance of the estimation steadily reduces with M, as
shown in Fig. 14. In other words, a higher precision of the
arithmetic used in orbit generation and greater samples
for the subsequent control parameter estimation, clearly
improves the results.
We conclude that the inaccuracies exposed above in
our method to recover the control parameter of maps of
F2, based on the order patterns of their coarse-grained
orbits (specifically, in form of Gray codes), are due to
n= 1−xn. Therefore, if x ∈ [0,1] is represented
the shortcomings of finite precision arithmetic and finite
statistical sampling, but are not inherent to the method
—as proved with the symmetric tent map.
102
104
106
M
108
1010
10−6
10−5
10−4
10−3
10−2
10−1
Error of the mean rate value with respect to λ/(2−λ)
FIG. 13: Dependency of the error in the estimation of the
rate of occurrences of the order pattern #0 with respect to
the length of the orbits.
102
104
106
M
108
1010
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Variance
FIG. 14: Variance of the error in the estimation of the rate
of occurrences of the order pattern #0 with respect to the
length of the orbits.
VII.CONCLUSIONS
In this paper it was shown how to rebuild the λ-DFs
of order patterns from Gray codes, the scope being the
estimation of the parameter λ. Gray codes are the 0-1
sequences that result from the symbolic dynamic of uni-
modal maps with respect to the left-right partition of the
state space introduced by the critical point. We have an-
alyzed the λ-DFs of the order patterns of two unimodal
parametric maps: the logistic map (as representative of
the subclass F1) and the skew tent map (as represen-
tative of the subclass F2). In the case of the logistic
Page 10
10
map, it turns out that this technique can hardly deliver,
on account of the complex and many-to-one relation be-
tween λ and those λ-DFs. On the contrary, this relation-
ship is simple, one-to-one, and analytically known for
π = [0,1,...,L−1] in the case of the skew tent map. Our
method improves previous proposals for parameter esti-
mation of unimodal maps in that a knowledge of the crit-
ical point value is not needed. However, it demands high
computational precision and large amounts of data; in
this regard, we recommend the use of extended-precision
libraries for good estimations. In the ideal case of arbi-
trarily high precision, the estimated value of the control
parameter is arbitrarily close to the real one.
Acknowledgments
The work described in this paper was supported
by Ministerio de Educaci´ on y Ciencia of Spain, re-
search grant SEG2004-02418, CDTI, Ministerio de In-
dustria, Turismo y Comercio of Spain in collaboration
with Telef´ onica I+D, Project SEGUR@ with reference
CENIT-2007 2004, CDTI, Ministerio de Industria, Tur-
ismo y Comercio of Spain in collaboration with SAP,
project HESPERIA (CENIT 2006-2009), and Ministe-
rio de Ciencia e Innovaci´ on of Spain in collaboration,
project CUCO (MTM2008-02194).
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