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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.

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2. The control parameter is the value of the critical

point, i.e., xc= λ. This leads to a new subclass of

maps F2.

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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

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# 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

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f(k)

λ(x)

x

(a)λ = 0.3

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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

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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

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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

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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-

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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

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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].

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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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