International Journal of Modern Nonlinear Theory and Application, 2012, 1, 14-31
http://dx.doi.org/10.4236/ijmnta.2012.11003 Published Online March 2012 (http://www.SciRP.org/journal/ijmnta)
Harmonic Analysis in Discrete Dynamical Systems
Gerardo Pastor1*, Miguel Romera1, Amalia Beatriz Orue1, Agustin Martin1,
Marius F. Danca2, Fausto Montoya1
1Instituto de Seguridad de la Inform ación, CSIC, Madrid, Spain
2Romanian Institute of Science and Technology, Cluj-Napoca, Romania
Email: *gerardo@iec.csic.es
Received February 7, 2012; revised March 9, 2012; accepted March 19 2012
ABSTRACT
In this paper we review several contributions made in the field of discrete dynamical systems, inspired by harmonic
analysis. Within discrete dynamical systems, we focus exclusively on quadratic maps, both one-dimensional (1D) and
two-dimensional (2D), since these maps are the most widely used by experimental scientists. We first review the appli-
cations in 1D quadratic maps, in particular the harmonics and antiharmonics introduced by Metropolis, Stein and Stein
(MSS). The MSS harmonics of a periodic orbit calculate the symbolic sequences of the period doubling cascade of the
orbit. Based on MSS harmonics, Pastor, Romera and Montoya (PRM) introduced the PRM harmonics, which allow to
calculate the structure of a 1D quadratic map. Likewise, we review the applications in 2D quadratic maps. In this case
both MSS harmonics and PRM harmonics deal with external arguments instead of with symbolic sequences. Finally, we
review pseudoharmonics and pseudoantiharmonics, which enable new interesting applications.
Keywords: Harmonic Analysis; Discrete Dynamical Systems; MSS Harmonics; PRM Harmonics; Pseudoharmonics
1. Introduction
In this paper we review a branch of harmonic analysis
applied to discrete dynamical systems. In general, har-
monic analysis has been widely used in experimental
applications, as in the field of signal processing. In the
same way, harmonic analysis applied to discrete dy-
namical systems can be a valuable tool for experimental
scientists studying nonlinear phenomena. This paper is
focused abov e all in showing some tools with interesting
applications in no nl i near phenome na.
At its inception, the harmonic analysis studies the rep-
resentation of a function as the superposition of basic
waves which, in physics, are called harmonics. Fourier
analysis and Fourier transforms are the two main branches
investigated in this field. The harmonic analysis is soon
generalized and in the past two centuries becomes, as
noted above, a wide subject with a large number of appli-
cations in diverse areas of experimental science.
In order to study nonlinear phenomena, experimental
scientists use dynamical systems, whether continuous or
discrete. In this review paper we only deal with discrete
dynamical systems and more specifically with quadratic
maps, above all one-dimensional (1D) quadratic maps
and two-dimensional (2D) quadratic maps, which are the
most commonly used.
The two most popular 1D quadratic maps are the lo-
gistic map
11
nnn
x
xx

2
and the real Mandelbrot
map 1nn
x
xc
. The logistic map [1-3] is widely
known among experimental scientists studying nonlinear
phenomena. Indeed, since Verhulst used it for the first
time in 1845 to study populatio n growth [1 ], it has served
to model a large number of phenomena. The real Man-
delbrot map is the intersection of the Mandelbro t set [4-6]
and the real axis. All the 1D quadratic maps are topo-
logically conjugate [7-9]. Therefore, we can use one of
them to study the others.
The most popular 2D quadratic map is, without any
doubt, the Mandelbrot set, which is the most representa-
tive paradigm of chaos. The Mandelbrot set can be de-
fined as
:0 as
k
c
Mc fk

C, where
0
k
c
f is the k-iteration of the complex polynomial
function depending on the parameter c,

2
c
f
zzc
0, z
and c complex, for the initial value . In the same
way as we use the Mandelbrot set to study the complex
case, to study the 1D case we normally use the real
Mandelbrot map [10-12] (likewise we could have used
the logistic map), that can be defined again as
z
:0 as
k
rc
Mc fk

C
, but where now
0fk
c is the k-iteration of the 1D polynomial function
depending on the parameter c,

2
c
xxc, x and c
real, for the initial value . In this real Mandelbrot
map there are several kinds of points according to the
0x
*Corresponding a uthor.
C
opyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL. 15
multiplier value


d
f
p
c
x
x
df xx
[13]. A similar
definition of the multiplier λ can be given for the Man-
delbrot set complex case, as can be seen for example in
[6], where c is defined for the parameter values of the
Mandelbrot set, which are complex values, and not only
for the real values of the real segment 21c 4.
In both cases, the real case and the complex one, when
1
1
one has hyperbolic points. The connected com-
ponents of the c-values set for which converges
to a k-cycle are periodic hyperbolic components (period i c
HCs), or simply HCs [6]. These periodic HCs verify

0
k
c
f
, which means they are stable (if 0
they are
superstable). A HC is a cardioid or a disc for the complex
case (2D hyperbolic components) and a segment for the
real case (1D hyperbolic components). Therefore, we can
speak indistinctly of periodic orbits (superstable periodic
orbit if 0
) or hyperbolic components, although in
1D quadratic maps we normally speak of periodic orbits,
and in 2D quadratic maps of HCs. The variety of names
is due to the fact that this paper is a review of several
papers. There are also points where 1
which means
they are unstable. These last points are, in addition,
preperiodic and they have been later called Misiurewicz
points [14-18].
When 1
one has non-hyperbolic points. These
points correspond to tangent bifurcation points (or cusp
points, where, in the case of a 2D quadratic map, a cardi-
oid-like component is born) and to pitchfork bifurcation
points (or root points, where, in the case of a 2D quad-
ratic map, a disc-like component is born).
Therefore in both cases, 1D and 2D quadratic maps,
the two most representative elements are HCs (remember
that they are more commonly called periodic orbits in 1D
quadratic maps) and Misiurewicz points. There are many
ways to identify both HCs and Misiurewicz points. For
example, we can recognize a HC by means of its period,
and a Misiurewicz point by its preperiod and period.
However, normally a lot of HCs have the same period,
and a lot of Misiurewicz points have the same preperiod
and period; hence, this way of naming them is not uni-
vocal. We are interested in names of HCs and Misi-
urewicz points that can identify them univocally. When
the names identify them univocally, we denominate them
identifiers. As we shall see later, in 1D quadratic maps
the identifiers we use are the symbolic sequences [19],
which are sequences of the type CXX (X can be a L
for left, or a R for right). These sequences show the
symbolic dynamics of the critical point in the map under
consideration. Unfortunately, symbolic sequences can
not be used as identifiers in 2D quadratic maps because
two different HCs can have the same symbolic sequence.
Therefore, to identify a HC (or a Misiurewicz point) in a
2D quadratic map we normally use the external argu-
ments (EAs) associated to the external rays of Douady
and Hubbard [15,20,21] that land in the cusp/root points
of the cardioids/discs (or in the Misiurewicz points).
These EAs are given as rational numbers with odd de-
nominator in the case of hyperbolic components, and
with even denominator in the case of Misiurewicz points.
These rational numbers can also be given as their binary
expansions [22], which are the most commonly used, and
the only ones used here.
As we shall see later, Metropolis, Stein and Stein
(MSS) [23] used a variant of the harmonic analysis
within the field of discrete dynamical systems, specifi-
cally within a type of 1D quadratic maps, the logistic
map. While in the classical harmonic analysis a function
is the superposition of the infinity of its harmonics, the
harmonics of MSS (MSS harmonics) of the symbolic
sequence (pattern for MSS) of a superstable orbit calcu-
late the symbolic sequences of the period doubling cas-
cade of the original orbit. Therefore, the MSS harmonics
are a very valuable tool since, given the symbolic se-
quence of a superstable orbit (which characterizes the
whole HC), the symbolic sequences of the infinity of
orbits of its period doubling cascade can easily be calcu-
lated. Indeed, if we start from a period-p orbit (or HC),
the periods of the orbits calculated are 2p, 4p, 8p,
(doubling period cascade, always in the periodic region).
In Section 2.1 we shall see in more detail MSS harmonics
and, in addition, we also shall see MSS antiharmonics.
Based on MSS harmonics and MSS antiharmonics,
Pastor, Romera and Montoya (PRM) introduced in [12]
Fourier harmonics (F harmonics) and Fourier antihar-
monics (F antiharmonics), which in their subsequent pa-
pers were simply called harmonics and antiharmonics in
order to avoid confusion within the Fourier analysis.
Nevertheless, if we simply call them harmonics, they can
be confused with the MSS harmonics. Therefore, in this
review we have called them PRM harmonics. While
MSS harmonics were introduced by using the logistic
map, PRM harmonics were introduced by using the real
Mandelbrot map. These PRM harmonics and antihar-
monics are a powerful tool that can help us in both the
ordering of the periodic orbits of the chaotic region (and
not only those of the periodic region as in the case of the
MSS harmonics) and the calculation of symbolic se-
quences of these orbits. As we shall see in more detail in
Section 2.2, given the symbolic sequence of a periodic
orbit, the PRM harmonics of this orbit are the symbolic
sequences of the infinity of last appearance periodic or-
bits of the chaotic band generated by such an orbit.
As we have just said, in Sections 2.1 and 2.2 we in-
troduce the harmonics/antiharmonics of MSS, and the
harmonics/antiharmonics of PRM respectively, in both
cases when the identifiers of the periodic orbits are the
symbolic sequences corresponding to 1D quadratic maps.
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
16
In Sections 3.1 and 3.2 we shall see again MSS and PRM
harmonics/antiharmonics but now in 2D quadratic maps;
that is, when the identifiers are EAs.
When we are working in the chaotic region of 2D
quadratic maps out of the period doubling cascade and
out of the chaotic bands, harmonics and antiharmonics
have to be generalized. That is what we do in Section 4,
where pseudoharmonics and pseudoantiharmonics are
introduced. These two new tools will allow new order-
ings and new calculations in this chaotic region.
2. Harmonics in 1D Quadratic Maps
In this section on 1D quadratic maps we first review the
MSS harmonics/antiharmonics, which were introduced
by MSS [23]. Given the symbolic sequence of a periodic
orbit, MSS harmonics calculate the symbolic sequences
of the period doubling cascade of that periodic orbit,
placed in the periodic region. Finally, we review the
PRM harmonics/antiharmonics [12]. Given the symbolic
sequence of a periodic orbit (HC), PRM harmonics cal-
culate the symbolic sequences of the last appearance pe-
riodic orbits (or last appearance HCs) of that periodic
orbit, placed in the chaotic region. Let us see both cases
in Sections 2.1 and 2.2.
2.1. MSS Harmonics
The symbolic dynamics is introdu ced by Morse and Hed-
lung in 1938 [24]. According to Hao and Zhen [25],
based on this theory, Metropolis, Stein and Stein [23]
develop the applied symbolic dynamics to the case of
one-dimensional unimodal maps, which is simpler and
very useful. Applied symbolic dynamics used in the pre-
sent paper is based on the paper of MSS and on the reci-
pes of Schroeder [26].
The symbolic dynamics is based on the fact that some-
times it is not necessary to know the values of the itera-
tion but it is enough to know if these values are on the
left (L) or are on the right (R) of the critical point (C).
The sequence of symbols CXXX (X is an L for left, or
a R for right) is called symbolic sequence, or pattern.
There are two types of 1D quadratic maps, rightward
maps, R-maps, and leftward maps, L-maps [11,12]. The
most representative R-map is the logistic map,
n
11
nn
x
xx

, and the most representative L-map is
the real Mandelbrot map, 2
1
nn
x
xc
. In the logistic
map the critical point is a maximum, whereas in the real
Mandelbrot map the critical point is a minimum. As said
before, all the 1D quadratic maps are topologically con-
jugate [7-9], therefore the logistic map and the real Man-
delbrot map have equivalent symbolic dynamics, and the
symbolic sequences of one of them can be obtained by
interchanging Rs and Ls from the other one.
MSS use the logistic map, an R-map, therefore the
R-parity, which is the canonical parity of a R-map, has to
be applied. The symbolic sequence of a periodic orbit of a
R-map has even R-parity if the number of Rs is even, and
it has odd R-parity if the number of Rs is odd [11,12]. Let
us see now the definition of harmonic introduced by MSS.
Let P be the pattern of a superstable orbit of the logistic
map. The first MSS harmonic of P,


1
MSS
H
P, is formed
by appending P to itself and changing the second C to R
(or L) if the R-parity of P is even (or odd). The second
MSS harmonic of P,


2
MSS
H
P, is formed by appending


1
MSS
H
P to itself and changing the second C to R (or L)
if the R-parity of


1
MSS
H
P is even (or odd). And so on.
The change from C to R or L obeys the relo rule (R if
Even and L if Odd), which is the canonical rule of a
R-map, and a useful mnemonic rule. As mentioned in the
introduction, the periods of the successive MSS harmon-
ics of a pattern P of period p are: 2p, 4p, 8p, , which
correspond to the periods of the patterns of the period
doubling cascade of P.
Example:
We start from the period-1 superstable orbit whose
pattern is C. To find the patterns of the period doubling
cascade of C we have to obtain the successive MSS har-
monics of C by applying the relo rule. To obtain the first
MSS harmonic of C we append C to C (CC) and we
change the second C to R because the R-parity of C is
even. To obtain the second MSS harmonic of CR we
append CR to CR (CRCR) and we change the second C
to L because the R-parity of CR is odd. And so on. The
results up to the fifth MSS harmonic, which correspond
to the 20, 21, 22, 23, 24 and 25 periodic orbits of the period
doubling cascade, are:


0CC
MSS
H
, , ,


1CCR
MSS
H


2C CRLR
MSS
H


33
CCRLR LR
MSS
H,


433
CCRLR LRLRLR LR
MSS
H,


533333
CCRLRLRLRLR LR LR LRLRLR LR
MSS
H

i
.
Note that, when i = 0,
M
SS corresponds to the trivial
case of the starting point, and only when
H1, 2,i
,
()i
M
SS
H are the first, second, MSS harmonics, respec-
tively.
Let us see now the definition of antiharmonics, also
introduced by MSS.
Let P be the pattern of a superstable orbit of the logis-
tic map. The first MSS antiharmonic of P,


1
MSS
A
P, is
formed by appending P to itself and changin g the second
C to L (or R) if the R-parity of P is even (or odd). The
second MSS antiharmonic of P,


2
MSS
A
P, is formed by
appending


1
MSS
A
P to itself and changing the second C
to L (or R) if the R-parity of


1
MSS
A
P is even (or odd).
And so on.
As can be seen, in this case the mnemonic rule is the
lero rule (L if Even and R if Odd), which is the antican-
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
Copyright © 2012 SciRes. IJMNTA
17
onical rule of a R-map. As in the case of the MSS har-
monics, the periods of the successive MSS antiharmonics
of a pattern P of period p are: 2p, 4p, 8p, Antihar-
monics seem to have no interest because they do not
correspond to any possible periodic orbit. However, this
is not so, as we shall see later.
of the symbolic sequences or patterns of superstable or-
bits. However, in the same way as the Sharkovsky or-
dering only treats a part of the total set of the superstable
orbits, the first appearance superstable orbits, PRM only
treat another part of this set, the last appearance super-
stable orbits. On the other hand, while MSS or Shark-
ovsky use the logistic map, a R-map, PRM use the real
Mandelbrot map, a L-map, which is the intersection of
the Mandelbrot set with the real axis.
Example:
If we start again from the period-1 superstable orbit
whose pattern is C, and we calculate up to the third MSS
antiharmonic by applying the lero rule, we obtain: From the introduction of PRM harmonics, PRM obtain
what they call the harmonic structure of a 1D quadratic
map [12], which results from the generation of all the
genes, i.e., the superstable orbits of the period doubling
cascade. This harmonic structure obtained from the genes
is a way of seeing the ordering that clearly shows the
connection between each period doubling cascade com-
ponent (gene) and the corresponding chaotic band.


0CC
MSS
A, ,


1CCL
MSS
A


23
CCL
MSS
A, ,


37
CCL
MSS
A
that indeed do not cor respond to any periodic orbit.
2.2. PRM Harmonics
In 1997 the PRM harmonics and PRM antiharmonics
were introduced to contribute to the ordering of 1D quad-
ratic maps [12]. The search of order in chaos, and more
specifically in 1D quadratic maps, was early carried out in
the well known works of Sharkovsky [27,28]. Shark-
ovsky’s theorem gives a clear ordering of the superstable
periodic orbits but only of orbits that appear by the first
time. This theorem states that the first appearance of the
periodic orbits of the parameter-dependent unimodal
maps are in the following universal ordering when the
parameter absolute value increases:
One can obtain all the structural patterns by starting
out only from the pattern C of the period-1 superstable
orbit. Beginning from this pattern C, all the patterns of
the period doubling cascade and the patterns of the last
appearance superstable orbits of the chaotic bands are
generated. One can clearly see that the origin of each
period- chaotic band n is the nth periodic orbit of
the period doubling cascade, with period , which is
the gene .
2n
G
B2n
n
1 2 4 8 ... 2k.9 2k.7 2k.5 2k.3 ... 2.9 2.7
2.5 2.3 ... 9 7 5 3 where th e s ymbo l mu s t b e
read as “precede to”.
The Sharkovsky theorem gives a clear ordering of the
first appearance superstable orbits (see Figure 1), but
without taking into account either the symbolic sequence
or the origin of each periodic orbit. On the contrary, the
outstanding work of MSS [23], which also deals with the
issue of ordering, uses both the symbolic sequence and
the pattern generation; however, it is difficult to see any
ordering there (see Figure 2, where we graphically show
the MSS superstable periodic orbit generation according
to the MSS theorem [23]). Figure 1. A sketch of the Sharkovsky theorem for the logis-
tic map,
11
nn
xxx

20p
n
. First appearance superstable
rbits for periods
As said before, PRM harmonics were introduced in
[12] to better understand the ordering and the generation
are shown.
o
Figure 2. A sketch of the successive application of the Metropolis, Stein, and Stein theorem in the logistic map

11
nn
xxx

n
for Symbolic sequences for per iods 10p. 6p
are shown.
G. PASTOR ET AL.
18
Let us consideal Ma
have repeated is an L-map. As we know from [12,17], a
pattern P has even L-parity if it has an even number of Ls,
and has odd L-parity otherwise. L-parity is a concept
similar to R-parity, introduced by MSS [23] for the logis-
tic map, a R-map. Now, the definition of PRM harmonics
[12,17] can be seen.
Let P be a pattern. The first PRM harmonic of P,
r the rendelbrot map, which as we


1
PRM
H
P
changing the
even (or


2
PRM
, is formed by appending P to itself and
second C to L (or R) if the L-parity of P is
odd). The second PRM harmonic of P,
H
P, is formed by appending P to


1
PRM
H
P
the L-parity
and
he second C to L (or R) if of changing t


1
PRM
H
P
As can be
is even (or odd), and so on .
seen, in this case the mnem
lero rule (L if Even and R if Odd), which is the canonical
rule for an L-map.
Application: The harmonic structure of 1D quadratic
maps
Let us see how to generate the chaotic bands in the real
Mandelbrot map by beginning just at the origin, i.e., at
the period-1 superstable orbit of symbolic sequence C.
For this purpose, we shall be assisted by Figure 3, where
in the upper parts we depict sketches of the PRM har-
monics where periods and symbolic sequences are shown
and, in the lower parts, we depict the corresponding
Mandelbrot set antenna zones by means of the escape
lines method [29]. Obviously, since we are in the 1D
case, in these lower parts only the intersection of these
figures with the real axis make sense, although we use
the whole figure in order to better “see” the periodic or-
bits. Let us note that, in the upper parts, symbolic se-
quences corresponding to cardioids are only depicted
with black half filled circles, while symbolic sequences
corresponding to discs are depicted with black circles. In
Figure 3(a) we show the PRM harmonics of C obtained
in accordance with the lero rule. To form the first PRM
harmonic of C we add a C to the C, i.e., we write CC and
we change the second C into an L, since the L-parity of
C is even. Therefore, the first PRM harmonic of C is
. To form the second PRM harmonic of
to the first one, i.e., we write CLC and we
nd C into a R, since the L-parity of CL is
ain
the we obtain that the
third and fourth PRM harmonics of C are
and . As is well
], CLR, CLR (CRL, CRL2,
bolic se-
last appeara orbits. As it
n [12,M har-
cacase the
the preperiod
onic rule is the


1CCL
PRM
H
C we add a C
change the seco
odd, and we ob t
By applying


2CCLR
PRM
H.
same procedure,


32
CCLR
PRM
H
known [12,26
CRL3, , in the l
quences of the
was already show
monics of a pattern
limit


C
PRM
H is
1 and period 1, 1,


43
CCLR
PRM
H
2, CLR3,
ogistic map) are the sym
nce superstable
30], the limit of the PR
n be calculated. In this
Misiurewicz point with
1
M
, wnce is (CL)R
(if C account, the preperiod is 2, and we
have
hose symbolic seque
is taken into
2,1
M
, as can be seen in [12]). This point is the tip
(C) [1 whose parameter value is
As seen in the figure, we start from the period-1 su-
persta C placed in the pehe first
PRMonic of C is the period-2 sble orbit of
the p doubling cascade. All the oarmon-
uperstable riod-20
chaotic baand are the last appeaperstable
orbis . If we consider th0 supersta-
gene , then the PRrmonics of
the generate period-20band
Howe 21able orbit n th
riodrated. Let hap
whendnew gene
Let’ at 3(b) show
harm
2,30],
ble orbit
harm
eriod
ics of C are s
nd
its of th
ble orbit C as
ne G
ver, a
ic regio
this or
onics of
2c .
riodic region. T
upersta
ther PRM h
orbits placed in the pe
rance su
e period-2
M ha
chaotic
placed i
us see what
1
G.
where we
0
B
band
a
ge
period-
is also g
bit is use
0
G
the
superst
ene
as a
Figure
0
n
s now look
0
B.
e pe-
pens
the
1CLG
. To form its PRonics we
add Chone and we chathe second C
into ordance with thrule. So, we
obtain tat theond, third, anth PRM har-
m
,
, and .
The limit of these harmonics
M harm
nge
e lero
d four
CLRL

1CLHG
is the Misiurewicz point
L to t
a L or a R,
h
onics of

1
PRM

3
PRM
e previous
in acc
first, sec
are
1
G

1CLRLHG,


23
1
PRM
HG

5
1CLRLHG

47
RL
PRM
1
CLR L
[16,17], that separates the
and od-21 cha
generom CL
whic ponds to t
periling casca
the M harm
period-21
are t appearan
Agaiwe consider C
PRMonics of th
chaotic ba1
B (and
now Figure
12,
mM
the peri
rated f
h corres
od doub
other PR
placed in the
he last
n, if
harm
nd
Let us see
and fo

2
HG
, placed in
period-2
otic band
is the pattern
he secon
de of th
onics of C
chaotic b
ce supe
L as a gene
e gene
a new
3(c)
harmonics of the genee first, second,
third,urth harm
89 012c
0c band 0
B
1
B. Tt harmonic
C period-22,
supeorbit of the
dn of C. All
arrstable orbits
nd in addition ,
of this band.
Gve that the
gee period-21
2
G).
ere the PRM
1.543 6
chaoti
he firs
LRL of
rstable
ic regio
e supe
1
B and,
orbits
1, we ha
nerate th
CLRL
we show
. Th
d
e perio
L
a
rstable
1
ne
wh
G
ge
2CLRLG
onics of 2
G are


13
CLRLRL
PRM ,

3
23
CLRL
2RL
PRM
HG ,


5
33
2
HG


7
43
CLRLRL
PRM , and 2
HG
the periodic region and the
the period-2
is the
CLRLRL
PRM ,
the first being a new gene in
others superstable orbits placed in 2 chaotic
band . The limit of these harmonics Misiurewicz
point
2
B

12
24,2
CLRL LRmM
1
[16,17] which separates
chaotic band 2 chaotic
band . Therefore the PRM harmgene
period-22new gene,
Figure 3(d), show
PRMonics of the previous p(23) gene
the period-2
2
B
generate the
CLRL
harm
1
B
chaotic band
, of the next chaotic band.
and the period-2
onics of the
2, and a
ere we
eriod-8
2
G
the
B
wh
3
3RLG
Finally, let us see
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL. 19
stable orbits of the period doublincascade iFigure 3. A sketch of the PRM harmonics of the first four sup
e PRM harmonic gener ation of last appearan
chaotic band B0; (b) period-2 chaotic band B1; (c) period-4 ch
3
3CLRL RLG. A en
erg n the real Man-
delbrot map. Thce superstable patter ns of ctic bands is shown. (a) Pe-1
aotic band B2; (d) period-8 chaotic band B3.
new period-16 ge in the periodic
ted. The limit
a the Miu
ng cascad
hao riod
reg
of these superst
ion, 4
G, and the last appearance superstable orbits of
the period-23 chaotic band 3
B are genera
ble orbitsisrewicz point

1
38,4
mM [16,17] that separates the period-22 chaotic
band 2
B and the period-23 chaotic band 3
B.
Generalizing, the PRM harmonics of the gene n
G
generate the last appearance superstable orbits of the
period-2n chaotic band n
B, and a new gene of the period
doublie, the gene 1n
G
is
. Likewise,


P
RM n
H
G
is a Misiurewicz point 1
2,2
nn
n
mM
[10], a primary
separator (or band-merging point) of the chaotic bands
1n
B
and n
B.
This double procedure (periodic orbits of the period
cascade generatiodoubling chaotic band generation)
contin , periodic orbits of the
n and
ues ifinitely and both
t].
of thrhe set of the
nde
e co
period doubling cascade onhe right and chaotic bands
on the left, meet in the Myrberg-Feigenbaum point [31
Every pattern of the period doubling cascade is the
responding chaotic band. Tgene
PRM harmonics of all the genes is what we call the har-
monic structure and is schematically shown in Figure 4.
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
20
Figure 4. A sketch of the harmonic structure of the real Mandelbrot map. Each period doubling cascade superstable orbit is a
gene Gn. The PRM harmonics of these genes generate the corresponding chaotic band Bn and the Misiurewicz point
that separates the chaotic bands Bn–1 and Bn.
The patterns of the harmonic structure are called struc-
tural patterns and all of them are PRM harmonics.
In Figure 4 we can see the periodic region and the
chaotic region separated by the Myrberg-Feigenbaum
point. Likewise, the chaotic region is divided in an infin-
ity of chaotic bands B, separated by Misiurewicz points
called separators, mn, . Each structural pattern
of each chaotic band and each separ ator is determined by
starting with the only datum of the pattern C, and by ap-
As can be seen in
gistic map, a R-
-m t
rightward ref, we have
to
rst Ponic of P
,1
22
nn
n
mM
n0n
plying the successive harmonics to the successive genes,
what shows the power as calculation tool of the PRM
harmonics. detail in [12], if we deal with the lo-
map, instead of with the real Mandelbrot
map, a Lap, the result is equivalent to what is shown in
Figure 4. However, we haveo take into account that in
this case, the canonical direction of the logistic map is the
direction, the canonical parity is the R-parity
and the canonical rule is the relo rule (theore
interchange Ls and Rs).
To finish this section, we shall see the PRM antihar-
monics.
Let P be a pattern. The fiRM antiharm,


1
PRM
A
P, is formed by ap itself and chang- pending P to
ing the second C to R (or L) if the L-parity of P is even
(or odd). The second PRM antiharmonic of P,


2
PRM
A
P,
is formed by appending P to


1
PRM
A
P
-parity of and
second C to R (or L) if the Lchanging the


1
PRM
A
P is even
M antihar-
never
t, although they
nt role in
p, to obtain
(or odd). A nd s o on.
As in the case of MSS an
monics are also a purely
correspond to a periodic orbi
have no real existence, the
ics. eith the lo
an
th
tiharm
form
t either. B
y play an
gistic map, a R
onics, PR
al construction and
u
importa
-ma
some cases, as we shall see later. However, in the case of
the structural patterns that we have treated here only
PRM harmonics are present.
We are dealing with the real Mandelbrot map, a L-map,
and we have to apply the relo rule to obtain antiharmon-
If we wre w
tiharmonics we would have to apply the lero rule; that
is, just the opposite than in the case of harmonics.
As said before, there are two types of 1D quadratic
maps, whose canonical direction are rightward for R-maps
and leftward for L-maps. The canonical rule of a R-map is
e relo rule, and the canonical rule of a L-map is the lero
rule. Harmonics (of MSS or PRM) are obtained by apply-
ing the canonical rules, and they grow in the canonical
direction. Likewise, antiharmonics (of MSS or PRM) are
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL. 21
obtained by applying the anticanonical rules, and they
grow in the anticanonical direction.
3. Harmonics in 2D Quadratic Maps
In the same way as in Section 2 we reviewed harmonics
(both MSS harmonics and PRM harmonics) in 1D quad-
ratic maps, in this section we review both types of har-
monics in 2D quadratic maps. The main difference be-
tween both cases is that the identifiers are the symbolic
sequences for the 1D case, and the EAs (which we only
use here in the binary expansion form) for the 2D case.
However, in both cases, the MSS harmonics calculate
the identifiers of the period doubling cascade, placed in
the periodic region, and the PRM harmonics calculate the
identifiers of the last appearance HCs (LAHCs), placed
in the chaotic region. As said before, the 2D quadratic
map used here is the Mandelbrot set, which can be seen
in Figure 5(a). Let us see next MSS harmonics in Sec-
tion 3.1 and PRM harmonics in Section 3.2.
3.1. MSS Harmonics
For the 2D case, when we start from a period-p HC and
we progress through its period doubling cascade, we find
discs whose periods are 21·p, 22·p, 23·p, ... In the same
way as MSS introduced the concept of harmonics in 1D
y itself an
bviously we
unimodal maps [23], by extension we call MSS harmon-
ics of a HC of the Mandelbrot set to the set constituted
bd all the discs of its period-doubling cascade
refer to their identifier). (oLet

12
.,.aa be the two EAs of a HC. By taking into
account that the EAs of the period-2 disc are
.01, .10
(see Figure 5(a)), it is easy to obtain the EAs of a MSS
harmonic of

12
.,.aa from the tuning algorithm of
Douady and Hubbard [21,32]. Therefore, we can define
the MSS harmonics as follows:
Let

12
.,.aa be the two EAs of a HC. The succes-
sive MSS harmonics of the HC are given by:


012 12
.,..,.
MSS
H
aa aa,


1121221
.,..,.
MSS
H
aaaaaa,

21 21221 2112
.,. .,.
MSS
H
aa aaaaaaaa

,
31 21221211221121221
.,. .,.
MSS
H
aaaa aaa aaaa aaa aaa a, …
12
.,.
MSS
H
aa
are the EAs of the Myrberg-Feigenbaum
point. This notable point has neither binary nor rational
EAs.
Example 1
The EAs of the Mandelbrot set main cardioid are
.0, .1
of the s. By applying the previous expressions, the EAs
uccessive discs of the period doubling cascade of
such a main cardioid can be calculated. The first MSS
harmonics, up to the fourth, are:


0.0, .1.0, .1
MSS
H,


1.0,.1 .01,.10
MSS
H,

2.0, .1.0110,.1001
MSS
H,

3.0, .1.01101001, .10010110
MSS
H and

4.0, .1.0110100110010110, .1001011001101001
MSS
H,
as can be seen in Figure 5( a) .
Example 2
If we start from the cardioid of any other midget, we
can also calculate the EAs of the discs of its period dou-
bling cascade. Look at Figure 5 where Figure 5(a) sws ho
the Mandelbrot set; Figure 5(b) shows a sketch of the
shrub (13) marked with the rectangle c in Figure 5(a);
and Figure 5(c) shows the shrub (13) that is a magnify-
cation of the mentioned rectangle c. Let

.00111, .01000
be the EAs of the period-5 cardioid placed in the branch
11 of the shrub (13) [33] (see firstly Figures 5(b) and
5(c)). Figure 6(a) shows a magnification of the branch 11
where this period-5 representative can be better observed,
and Figure 6(b) shows an additional man of such
a period-5 representative. Thegnificatio
EAs of the successive discs
of the period doubling cascade of such a period-5 cardioid
can be calculated. Calculating the MSS harmonics, up to
the third, we obtain:


0.00111, .01000.00111, .01000
MSS
H,

1.00111, .01000.0011101000, .0100000111
MSS , H

2.00111,.01000.00111010000100
MSS
H

00 ,
0111, .01000001110011101000

3.0011101000010
.00111, .01000.010 000011100111
MSS
H
as can be seen in Figure 6(b).
Antiharmonics of MSS seem again to have no interest,
as we can deduce next from th
000
eir definition:
011101000001110011101000,
0100000111010000100000111
,,
Let

12
.,.aa be the two EAs of a HC. The success-
sive MSS antiharmonics of the HC are given by:


0.,. .,.
12 12
MSS
A
aa aa,



1
.,..,.


212111122 2 2
.,. .,.
MSS
A
aa aaaaaaaa,
Therefore, all of them are the same, which is the start-
ing HC.
3.2. PRM Harmonics
12 1122MSS
A
aa aaaa, L et us see now the PRM harmonics of a HC in this 2D case.
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
22
Figure 5. (a) Mandot set with the first fourelbr external argu
rnal as. Shrub (1/3) is framed in rectangle c. (b) Sk
in (a). (c) Shrub (1/3) that is a magnification of the rec
ments of the period doubling cascade, and other significant ex-
te rgumentetch o
tangle c markf the shrub (1/3) corresponding to the rectangle c marked
ed in (a).
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL. 23
Figure 6. (a) Magnification of the branch 11 of the shrub (1/3) shown in Figures 5 (b) and (c), where the period-5 representa-
tive can be better observed. (b) Magnification of such a period-5 representative shown in the rectangle b marked in (a), with
some of its important external arguments.
Let

12
.,.aa
expansion
be the EAs of a HC given in form of
binary[22]. The EAs of the order i PRM har-
monics of

12
.,.aa are given by:


121222211 1
.,. .,..
i
PRM
ii
H
aaaaaa aaaa





   (1)
When , Equation (1) calculates a se-
quence oEquati on (1) becomes:
0,1, 2, 3,i
f HCs. When i,

 
121222211 11221
.,..,.. ,.
PRM
H
aaaaaaaaa aaaaa








(2)
two preperiodic arguments, and therefore Misiurewicz
points.
In the 1D case, we obtained the harmonic structure
through repeated application of the PRM harmonics.
Similarly, in the 2D case Equation (1), applied to a
given HC when 0,1, 2, 3,i
f
inal HC. Indeed
). By applying Eq
M harmonics of G
se that normally n
, calculates a sequence of
HCs which are the LAHCs othe corresponding chaotic
band of the orig, let us analyze, as an ex-
ample, the main cardioid that we consider as a gene
(see Figure 7uation (1) we obtain
successive PR. For we ob
, a trivial cat willen into
count, for
0
G
the
tain
ac-
0
o0i
be tak
0
G1i
we obtain
2i we o
0. If we apply n
isiurewicz point
a new gene
b
o
1
G in
e
w Equation (
the
regitain thLAHCs of
chaotic band 2) to
ain t
peri-
the
0
G,
odic
we obt
on, and for
B
he M1,1
M
, which
is, is th
xtrem
e up-
beper ee of the chaotic band 0
B. That0
G can
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
24
Figure 7. The neighbourhood of the main antenna of the Mandelbrot set with three chaotic bands, B0, B1 and B2, showin g the
EAs of their first LAHCs. Likewise, the main cardioid G0 and the discs of the period doubling cascade, G1, G2, ···, showing
their EAs.
considered the gene or generator of the chaotic band
Likewise, by applying now Equation (1) to the new
, we obtain, after the trivial case of , first a new
in the periodic region, and tLAHCs o
nd . If we apply nowtion (2
ain thsiurewicz point
0
B.
gene
f
) to
1
G
gene
1
G
0i
hen the
Equa
2,1
2
G
the chaotic ba
, we obt1
B
e Mi
M
, wh
. That is,
th
1) to
ew gen
ich
e chaotic band
ered thgene or generatoe chaot
. Inneral, bying Equat a
, after thial case, first a ne
is the
can
band
n
G
1
upper extrem
consid
ge
tain
of the
e
y appl
e triv
1
r of
on (
B
i
1
G
ic
gene
n
G
be
1
B
we ob
in
ob-
ppe
be
n
B.
oni
r
c
th
band
tain
extre
consi
e peric regid then
. If we a now Equa
isiur oint
the ge
Wh we hast seen
structure of the 1D case. However, since now we are in
the 2D case, we only calculate the structure of the chaotic
bands of the cardioid considered. Let us see two exam-
ples, the first one applied to the main cardioid (see Fig-
ure 7), and the second one to the period-5 midget of the
branch 11 of the shrub
odi
n
B
the M
m
dered
at
on, an
pply
ewicz p
e of the chaotic band
gene or
ve ju
the LAHCs of the chaotic
to
ich
. That is,
e chaotic
lar to the h
tion (2)
1
h
i
n
G
is th
n
G
, we
e u
can
arm
2,2
nn
M
n
B
nerator of t
is sim
, wh
band
(13) in the chaotic region (see
Figure 6(b)).
Example 1
In Figure 7 we can see the neighbourhood of the main
antenna of the Mandelbrot set with three chaotic bands,
and , and also the main cardioid and
scs ofperiod doubling cascade, .
of tod-1 main cardioid, the g, are
0
B,
the d
The E
1
B
iAs
2
B
the
he peri
0
G
, 2
G
0
G
1
G
ene ,

0.0,.1G. If Equation (1) is applied to th is main car-
dioid,


.0,.1
i
PRM
H, for 0,1,2,3,i, the sequence
.0,.1 ,
.01, .10,
.011, .100,
.0111, .1000,
.01111, .10000,
is obtained. The values of this sequence (without taken
into account the first one) are: first the gene, and
then the LAHCs of the chaotic band . If no ap-
ply Equation (2), we obtain
1
G
w we
0
B


.0,.1 .01,.10
PRM
H, a
Misiurewicz point, ,101
mM
, which is the upper ex-
treme of consider now
0
B. Likewise, let us

1.01, .10.01,.10
PRM
H as a ne w gene
1.01, .10G.
By applying Equation (1) to, 1
G


.01, .10
i
PRM
H for
0,1, 2, 3,i
, we obtain the sequence
.01, .10,
.0110, .1001,
.011010, .100101,
0, .10010101, .0110101
1010, .1001010101,
which corresponds to (after the obvious ) first the
gene , and then the LAHCs of the chband
If no apply Equation (2) we obtain
.011010 ,
1
G
aotic
2
G
w we1
B.

.01, .10.0110,.1001
PRM
H, a Misiurewicz point,
,112
mM
, which is the upper extreme of B. Finally, we
can consider
1


1.01, .10.0110, .1001
PRM
H as a new
gene
2.0110, .1001G. By applying Equation (1) to
2
G,

.0110, .1001
i
PRM
H for 0,1, 2, 3,i, we ob-
tain the sequence
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL. 25

.0110, .1001,

.01101001, .10010110,

.011010011001, .100101100110,

.0110100110011001, .1001011001100110,

.01101001100110011001,.10010110011001100110.
After , these terms are first the gene and then
th of the chaotic band . By aping Equa-
tion (2ain
2
G
e LAHCs
) we obt
3
G,
ply
2
B


.0110, .1001.01101001,.10010110
PRM
H,
a Misiurewicz point, ,2
, which is the upper ex-
treme of B2. And so ch that, in a general case, Gn
generates the chaotic band Bn (not shown in the figure).
Example 2
Let us see the period-5 midget of the branch 11 of the
shrub(
24
mM
on, su
13)
th shown in Figure 6(b) which is a magnifica-
tion of e rectangle b shown in Figure 6(a). Let us con-
sider this period-5 cardioid as a gene whose EAs are

0.00111, .01000G.
If we apply Equation (1),




0.00111, .01000
ii
PRM PRM
HGH, for
we obtain the sequence
0,1, 2, 3,i,

.00111, .01000,

.0011101000, .0100000111,

.001110100001000, .010000011100111,
.00111010000100001000, .01000001110011100111,,
whose terms (for ) are the LAHCs of the chaotic
band . If now y Equation (2),
2i
we appl
0
B


.00111, .01000.0011101000,.0100000111
PRM
H,
a Misiurewicz point, 05,5
mM
, which is both the tip of
edget and the upper e
void probgure.
Antiharmonics of PRM neither seem to have any in-
terest, as in previous cases. Indeed, according t
definition:
th mixtreme of 0
B. As always, we
can do the same procedure in order to calculate the
LAHCs of the rest of the chaotic bands. We have not
done it to alems with the fi
o their
Let

12
.,.aa be thhe E
e EAs of a HC given in form of
binary expansions [22]. TAs of the order i PRM an-
tiharmonics of

.,.aa given by:
12
are

 
12 11112222 12
.,. .,..,.
i
PRM
ii
A
aaaaa aaaaaaa


  
Equation (3) b ecomes:
(3)
When i,

 
12 1111222212
.,. .,..,.
A
aa aaaaaaaa aa







 (4)
The result in both equ ations is the starting HC. However,
these concepts will be very useful in th e next section.
4. Pseudoharmonics and
Pseudoantiharmonics in the 2D Case
We have just seen that the PRM harmonics are a power-
ful tool for calculating some EAs in 2D quadratic maps.
However, we can go much further if we introduce an
extension of these calculation tools, which we simply call
pseudoharmonics and pseudoantiharmonics [35]. These
new tools are applied to the EAs of two HCs, as we can
see next.
4.1. Introduction of Pseudoharmonics and
Pseudoantiharmonics
The introduction of pseudoharmonics and pseudoanti-
harmonics can be seen in detail in [35]. Let us first in-
troduce pseudoharmonics. Let

12
.,.aa be the external
arguments of a HC, and let

12
.,.bb be the external
arguments of other HC that is related with the first one,
as will be seen later. The external arguments of the order
i pseudoharmonics of
12
.,.aa and

12
.,.bb are:


12121222211 1
.,. ;.,..,.
i
ii
PHaabbabbbab bb







  
(5)
When 0,1,2, 3,i
of HCs that we, Equation (5) calculates a se-
quence shall determinafterwards. When e
i, Equation (5) become s:



.,.abab



  
12121222211 1
.,. ;.,..,.PHaabbabbbab bb




12 21
(6)
Equation (6) calculates a pair of extern al arguments of
a Misiurewicz point. Later we shall analyze th is result in
every possible case.
Let us now introduce pseudoant iharmoni c s. Let

12
.,.aa be the external arguents of a C, and let mH
12
.,.bb be external arguments ofther HC, that is
related with the first one. The external arguments of the
order i pseudoantiharmcs of
e th o
oni

12
.,.aa and
12
.,.bb
are:
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
26


121211112222
.,. ;.,..,.
i
ii
PAaabbab bbabbb







 
(7)
When , Equation (7) calculates a se-
quence and not the starting HC as in the case of
Equatioe, although PRM antiharmonics
seemed tey have been useful in order to
introduonics. When , Equation
(7) becomes:
0,1, 2, 3,i
of HCs,
n (3). Therefor
o be of no use, th
ce pseudoantiharmi



121211112222
112 2
.,. ;.,..,.
.,.
PAaabbab bbabbb
aba b








 
(8)
Equation (8) calculates a pair of EAs of a Misiurewicz
point, and not the starting HC as in the case of Equation
late more EAs.
ne, h
av
(4). Again PRM antiharmonics are suitable to introduce
pseudoantiharmonics that, as we shall see, are very use-
l to calcufuPseudoharmonics and pseudoantiharmonics are, as has
been said, a generalization of PRM harmonics and PRM
atiharmonics in the 2D case; therefortese two last
ones he to be a particular case of the two first ones. In-
deed, when

12 12
.,..,.bb aa, Equations (5), (6), (7) and
(8) become Equations (1), (2), (3) an (4). Thus, in this
case, pseudoharmonics and pseudoantiharmonics have
become PRM harmonics and PRM antiharmonics. Let us
see it from other approach. Pseudoharmonics and pseu-
doantiharmonics are applied to two HCs whereas PRM
harmonics and PRM antiharmonics are applied to only
one HC. However, PRM harmonics and PRM antihar-
monics can be thought as pseudharmonics and pseudo-
antiharmonics where their two HCs are the same.
d
o
4.2. Some Considerations to Calculate
Pseudoharmonics and Pseudoantiharmonics
To apply what we have seen so far, we shall briefly re-
member some concepts that can be seen in detail in [35].
We call a descendant [35] of

12
.,.aa and
12
.,.bb
to any of the H
Cs obtained from Equations (5) and (7).
Lizone occupied by all the descendants cal-
cu
firs air
kewise, the
lated by using these equations is the zone of descen-
dants [35]. In other to calculate a descendant, given the
t p,

12
.,.aa, the second pair cannot be any
2
,.b.
Indeed,
1
.b

12
.,.bb has to be an ancestor [35] of

12
.,aa. .
Let us briefly see the ancestors of

.,.aa, which will
be usedation oonics and pseu-
12
in the calculf pseudoharm
doantiharmonics (descendants). The second pair
12
.,.bb is an ancestor of the first pair

12
.,.aa if
12
.,.aa is a descendant of

12
.,.bb and
12
.,.cc,
where
12
.,.cc itself is an ancestor of

12
.,.bb. Note
that any HC is an ancestor of itself. Note also that the
ancestor of a HC has lesser or equal period than the pe-
riod of such a HC. In order to determine the ancestors of
12
.,.aa, next we shall remember some points.
As known from [33,36], the shrub of the n-ary hyper-
bolic componen t 1
1
1
1
N
N
q
q
pp
, shrub(1
1
1
1
N
N
q
q
pp
), has n
subshrubs or chaotic bands, 1
N
SS
gene or , each one of them
generated by one HC called generator [33,35,36]
of the chaotic band or subshrub. The generator of the last
subshrub, , is the main cardioid 1
N
S, that is e first
1
component of the generation route of
th
11
1
1
N
N
q
q
. The
pp
generator of the last but one suub, , is the pri-
mary disc
bshr
1N
S
1
1
1
1
q, that is the se component of the cond
p
generation route of 1
1
1
1
N
N
q
q
pp
. Likewise, the generator
of S2N
, is the secondary disc 12
12
1qq
1pp
 , that is the
third component of the generation route of 11
11
N
N
q
q
,
pp
and so on. Finally, the gen subshrub, S erator of the first1,
is the (N-1)-ary disc 1
1
11
1
1N
pp
, om-
eneration roe of
N
q
q
that is the Nth c
ponent of the gut11
N
1
1
N
q
pp
. Hence, in
general, the generator of the subshrubis the (N-i)-
ary disc
q
i
S
1
1
1
1
N
i
N
i
q
q
pp
, that is the (N-i+1)th component
of the generation route of 1
1
1
1
N
N
q
q
pp
.
Let
12
.,.aa be the representative of a branch (whose
branch associated number is.
The first ancestor of 12 m
dd d [33]) of i
S
12
.,.aa is the generator
1
1
1
1
N
i
N
i
pp
q
q
i
S where
the br found. The second, third,
of the chaotic band or subshrub
anch is
ancestors of
12m
dd d
12
.,.aa are: 1
1
1
1
1
Ni
1
N
i


,
q
q
pp
2
1
1
1Ni
q
q
pp

12Ni
in the ge ne ra tion rout e of
, ,
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL. 27
1
1
1
1
N
N
q
q
pp
, up to 1
1
11
1
1N
N
q
q
pp
and 1
1
1
1
N
N
q
q
pp
, all
of them in the periodic region. Tlowing ancestors
of
he fol

12
.,.aa, now in the chaotic region, he repre-
sentatives of the branches,
can be seen, all the ancestors oe in
thbs
are t
. As
1
d, 12
dd ,
of the cha12 m
dd d
tic region ar
Si, andere is no one in the other subshru. What we
call ancestor route includes all the ordered ancestors of

12
.,.aa from the generator 1
1
1
1
N
i
N
i
q
q
pp
to
1
1
1
1
N
N
q
q
pp
in the peion, followed by the rep-
resentatives of the branches 1
d,
12
dd ,, 12m
dd d in
thion.
riodic reg
on
e chaotic reg
4.3. Zes of Descendants
As can be seen in detail in [35], depending on the ances-
tor

12
.,.bb we shall have a different zone of descen-
dants. Let us see the most important cas. es
1st case:

12
.,.bb is the first ancestor
If

12
.,.aa is the representative ch
of , the first ancestor of
of the bran
12 m i
dd dS

12
.,.aa,

12
.,.bb, is the neratorge 1
1
1
1
N
i
N
i
q
q
pp
of5]. In
this case,
i
S [3


12 12
.,. ;.,.
i
PHaabb


cathe rep-
resentatives of the brancheshat i
an nd so on. Ob-
viously, wh
lculates
. T
12 11
m
i
dd d s to
say, it successively calculates first the representative of
the branch 12m
dd d, and then the representatives of
the brches 12 1
m
dd d, 12 11
m
dd d, a
en i one has:


1 1221
.,.. ,.PHaaabab

that is the pair of external arguments the Misiurewicz
point placed in the upper extreme of the branches
12 11
m
dd d. Therefore,

2 12
;.,.bb

of

.,. ;.,..PHaabba b

12 121221
,.ab

calculates the upper extreme of the zone of descendants
of

.,.aa
12
and

12

.,.bb, and

12 12
.,. ;.,.
i
PHaab b


calculates de LAHCs in that
zone. The zonendants finishes in a
of desce tip or upper
extreme of ; therefore, this zone of descendants cov-
ers the whubshrub
2nd case:
i
S
ole s
i
S.

12
.,.bb is the second ancestor,
1
1
11
1
12
.,.aa be the representative of a branch
of . Since the first ancestor is
1Ni
N
i
pp

q
q
Let
12 m
dd di
S
1
1
1
1
N
i
N
i
q
q
pp
, that is the generator of i
S, the second
ancestor is 1
1
11
1Ni
1
N
i
pp
q
q
. Let

12
.,.bb be the external
arguments of this s econd ancestor. Then


12 12
.,. ;.,.PHaab b
and


12 12
.,. ;.,.PAaabb
respectively calculate the upper and lower extremes of
the zone of descendants of
12
.,.aa and

12
.,.bb. And,
in this case, the zone of descendants is the branch
12 m
dd d. On the other hand,


12
.,. ;.,.
i
PHaab b
12
and


12 12
.,. ;.,.
i
PAaab b
, 0,1, 2, 3,i,
respectively calculate the LAHCs i increasing and
decreasing directions of the brand.
3rd case:
n the
ch 12m
dd
12
.,.bb is the third, fourth, …, ancestor
(except for the last one).
Let
12
.,.aa be the representative of a branch
12 m
dd d of i
S. Its ancestors in the periodic region are:
1
1
1
1
N
i
N
i
q
q1
1
11
1
1Ni
N
i
q
q
pp


pp
,
,
2
1
12
1
1Ni
Ni
q
q
pp

, , 1
1
1
1
N
N
q
q
pp
,
ancestors in the c
fore, t
d its anhaotic region are the representa-
tives of the branches 1
d,
12
dd , , 12 m
dd d. There-
he third ancestor is 2
1
1q
12
1Ni
Ni
q
pp


if we are still
in the periodic region, and the representative of the
branch if we are not. Now,
1
d


12 12
.,. ;.,.PHaab b

and

12 12
.,. ;.,.PAaabb
respectively calculate the upper and lower extremes of
the zone of desce nda
nts of
12
.,.aa and

12
.,.bb, that
in her
hand,
this case is a sub-branche of dd d. On the ot
12 m


12 12
.,. ;.,.PHaab b
i
and


12 12
.,. ;.,.
i
PAaab b
, 0,1, 2, 3,i,
respectively calculate the LAHC
decre s in the increasing and
asing directions of this sub-branch of .
12 m
dd d
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
28

12 12
.,..,.bb aa. 4th case:
As was already seen, when

12 12
.,..,.bb aa Equa-
tions (5), (6), (7) and (8) become Equations (1), (2),
anence pseudoha
to an
ely the upper extreme and the LAHCs of
the chaotic band associated to any disc.
4.4. Applications
Let us see an example for each of the four previous cases.
Let us consider again the period-5 HC
(3)
d (4), hrmonics and pseudoantiharmon-
ics become PRM harmonics and PRM antiharmonics.
Therefore, in this case, by using pseudoharmonics and
pseudoantiharmonics one can calculate the two extremes
and the LAHCs of the midget associatedy cardioid,
and alternativ

12
, ..00111, .01000, that is the representative
the bran ch 11 of the sh rub (
.aa of
13). We help us with Figure
5, which shows such a shrub (13), and with Figure 6
whose part (a) is a magnification of the branch 11, and
whose part (b) is an additional magnification of the pe-
riod-5 representative. If one starts from

12
., ..00111, .01000aa, the possible values of

12
.,.bb are, as seen before:
.0,.1 and

.001, .010
in the periodic region, and

11, .0100 and .00

.00111, .01000 in 1
S, that is the only subshrub of the
chaotic region .
4.4.1. First Example
Let us begin with

12
.,. .0,.1bb , which corresponds
to the 1st case because

12
.,.bb is the generator of 1
S,
and therefore the first ancestor.


.00111, .01000;.0, .1.001,PH 

e Misiurewicz point 2,1
.010
is th
M
placed in the upper ex-
treme of the branch 111, ftip ( 13), that is the upper
extreme of the increasing part of the zone of descendants
of


.00111, .01000;.0, .1


, a clearly seen in s can be
Figures 5(b), 5(c), and 6(a).



.00111,.01000;.0, .1
i
PH 

calculates

.00111, .01000,

.001111, .010000,

..0100000 , ,
which arHCs in this increasing part of o
0011111,
e the LA the zonef
de he repre
thast one i
scendants, or, what is the same, tsentatives of
e branches 11, 111 and 1111 (the ls not shown
in the figure). In general, the increasing part of the zone of
descendants starts from
.,.aa, that is the repr
12
the branch dd d, and it reachesper
of its chaotic band by ng the s
12 dd d
only caich pseudoantiharmonics can not be used
esenta-
tive of the up
extreme followibranche
, . This is the
n
12 m
1
m, 12
dd
m
dd dse i12
wh11
m
d,
in order to calculate the decreasing part of the zone of de-
scendants of

.00111, .01000;.0, .1
.
4.4.2. Se
Le
cond Exam ple
t us go on with

12
.,..001,.010bb , that corresponds
to the 2nd case in w
2
.b is thncestor,
1
.,be second ahich
corresponding to 1
1
1
1
N
N
q
q
pp
, that in this case is the last
ancestor in the periodic region. The upper and lower ex-
tremes of tzone of descendants of he
.00111, .01000;.001, .010
are



.00111, .01000;.001, .010
0111010,.01000001 and
PH
.0



.001101,
.00111, .01000;.001, .010
100,.01000
PA
that the Misiurewicz points 5,1
M
an ared 4,1
M
, ex-
trembranch 11 (see Figures 5(c 6(a)). As
always,

es of the ) and

.00111, .01000;.001, .010
i
PH
and


.00111, .01000;.001, .010
i
PA
calculate the LAHCs in tpart
(
he increasing
.00111, .01000,
00001 , .00111010, .010
.00111010010, .01000001001,) and in the decreas-
ing part (
0,
.00111001, .01000010), .00111, .0100
.00111001001, .01000010010,
scendants of
) of the zone of de-

.00111, .01000;.001, .010
the branch 11.
4.4.3. Third Exampl
, in this case
e
Let us go on with

12
.,..0011, .0100bb (th3rd case e
in which
12
,. is .bbthe third ancestor).

.0011, .0100
is the represef the br firsntative oanch 1, i.e. thet ancestor
in the chaotic res
of the zone of degion 1
S. The upper and lower extrem
escendants of
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL. 29

1000 ; .0 are
.00111, .0011, .0100




.001110100,.010000011 and
.00111, .01000;.0011, .0100PH 



.00111, .01000;.0011, .0100
.0011100,.0100001,
PA 

that are the Misiurewicz points 5,2
M
and 3, 2
M
, which
are the e
Figure 6(a)). This sub-branch is ca
xtremes of a sub-branch of the branch 11 (see
lled

m112
s
ub dd d.
Again,


.00111, .01000;,.010PH

.0011 0
i and

1, .01000;.0011, .0100
easing part
.0011
i
PA
calculate the LAHCs in the incr
(

.00111, .01000,

.001110100, .010000011,

.0011101000100, .0100000110011,) and in the de-
creasing part (
.00111,.01000,
1,.010000100 ,
.00111001

001011, .0100001000100,) of the zone. 1100110 of
descendan s
ts of uch a sub-branch, m
112
s
udbd d
, although only pe
If we in a case
, in
both cases with periods 5, 9, 13, ri-
ods 5 ane figure.
where d 9 are shown in the wer

12
.,.bb is the fourth ancestlast or (but not the
one), then


12 12
.,. ;.,.PHaab b

and


.,. ;.,.PAaabb

wr and
12 12ould be the uppe

lower extremes of the zone of descendants of

12 1
.,. ;.aab
ichisiurewicz points
that are the ex of a sub-bra
2
,.b
, wh are the M
tremesnch

212 m
s
ubddd

m
of the previous sub-branch 112
s
ubd dd. And so on.
4.4.4. Fourth Example
Finally, let us see the nth case, that corresponds to

12
.,..00111, .01000bb , i.e. when

12 12
.,..,.bb aa. The upper and lower extremes of the
zonendante of descs of

.00111, .01000;.00111, .01000
are


.00111, .01000;.00111, .01000
.0011101000,.0100000111 and
PH 



.0011
.00111,.0 1000,


that are the tip a to
1, .01000;.00111, .01000PA
nd the cusp of the midget associated
12 12
.,..00111, .01000b, the first., .aa b one a
Misiurewicz point 5,5
M
and the second one the cusp of
the original HC

12
.., ..00111, .01000bb
6(b)). As alwa
12
, .aa
(see Figure ys,

.00111, .01000;.001, .010PH i
calculates the
LAHCs of the increasing part of the zodants ne of descen
(
.00111, .01000,
.0011101000, .0100000111,
.001110100001000, .010000011100111,
.00111010000100001000,1110011 ,). .0100000 110011
scendants e cusp of
5. Conclusions
dratic maps, both 1D
qups and 2 by experimental scientists, to
this paper.
vie aps,
in particul h
by MSS. Giverioe M
the symbolic sequence of this orbit calculate the sym-
bolic sequences of the period doubling cascade of the
orbit. Metropolis, Stein and introducedthe first
time this type of calculationand the concept of harmon-
n, Pasmera and
introduced the PRM harmonics. These type of harmonics
the calculation
band where
the original orbit is. Thus, if we start froiod-1 or-
bit, we can calculate the structure of a 1D quadratic map.
Both, MSS harmonics and PRM ha rede-
rs instead of
the external argument
nd the PRM ha
AHCs of its chaotic
Pseudoharmonics and pseudoantiharw are
an expansion of PRM harmonics and PRM antiharmoni
in the case of 2D quadratic maps, are a reviewed. The
d make possi-
This increasing part coincides with all the zone of de-
scendants because the decreasing part of the zone of de-
becomes a point, ththe original HC.
We review a branch of the harmonic analysis applied to
dynamical systems. We focus on discrete dynamical sys-
tems, and more specifically on qua
adratic maD quadratic maps, since these maps
are the most widely used
whom we have addressed
We first rew the applications in 1D quadratic m
ar thearmonics and antiharmonics introduced
n a pedic orbit, thSS harmonics of
Stein for
,
ics in dynamical systems.
Based on MSS harmoicstor, RoMontoya
expand the possibilities of s. So, given the
symbolic sequence of a periodic orbit, the PRM harmonics
of this orbit are the symbolic sequences of the infinity of
last appearance periodic orbits of the chaotic
m the pe
rmonics are
fined in order to can be used in 2D quadratic maps where
external arguments are used as identifie
symbolic sequences. Givens of a
cardioid, the MSS harmonics calculate the EAs of the
discs of its period doubling cascade, ar-
monics the EAs of its Lband
monics, cs
lsose
0
B.
hich
new calculation tools are notably more powerful than
PRM harmonics and PRM antiharmonics an
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
30
ble the calculation of LAHCs inw zones of descendants.
6.
on Neumann, “On Combination of Sto-
inistic Processes,” Bulletin of the
[3] R. M. May, “Simple M
Complicated Dynamics,”0
ne
Acknowledgments
This work was partially supported by Ministerio de Cien-
cia e Innovación (Spain) under grant TIN 2011-22668.
REFERENCES
[1] P.-F. Verhulst, “Recherches Mathématiques sur la Loi
d’Accroissement de la Population,” Mémoires de l’Aca-
démie Royale des Sciences et Belles-Lettres de Bruxelles,
Bruxelles, 1844.
[2] S. Ulam and J. v
chastic and Determ
American Mathematical Society, Vol. 53, No. 11, 1947, p.
1120.
athematical Models with Very
Nature, Vol. 261, 1976, pp. 459-
467. doi:10.1038/261459a
[4] B. B. Mandelbrot, “Fractal Aspects of the Iteration of
1zzz
 for Complex
and z,” Annals of the New
York Academy of Sciences, Vol. 357, 1980, pp. 249-259.
doi:10.1111/j.1749-6632.1980.tb29690.x
[5] B. B. Mandelbrot, “O of n the Quadratic Mapping
2
zz
 for Complex and Frac
Its μ Set, and Scaling,” D: Non
nomena, Vol. 7, 1983, pp
z: Thetal Structure
of Physica linear Phe-
. 224-239.
3)90128-8doi:10.1016/0167-2789(8
[6] B. Branner, M
posia in Appa
nd W. Thurston, “On Iterated Maps of the
Interval,” Dynamical Systems, Vol. 1342, 1988, pp. 465-
Fb0082847
“The andelbrot Set,” Proceedings of Sym-
lied Mthematics, Vol. 39, 1989, pp. 75-105.
[7] J. Milnor a
563. doi:10.1007/B
. Jürgens and D. S
New York, 1992, pp
ston, 1989, pp. 44-48.
omera and F. Montoya, “An Approach to
the Ordering of One-Dimensional Quadratic Maps,” Chaos,
l. 7, No.
[8] H.-O. Peitgen, Haupe, “Chaos and Frac-
tals,” Springer, . 569-574.
[9] R. L. Devaney, “An Introduction to Chaotic Dynamical
Systems,” Addison-Wesley, Bo
[10] G. Pastor, M. R
Solitons & Fractals, Vo4, 1996, pp. 565-584.
doi:10.1016/0960-0779(95)00071-2
[11] M. Romera, G. Pastor, J. C. Sanz-Martín and F. Montoya,
“Symbolic Sequences of One-Dimensional Quadratic
Maps,” Physica A: Statistica
tions, Vol. 256, No. 3-4, 1998, p
00083-1
l Mechanics and Its Applica-
p. 369-382.
doi:10.1016/S0378-4371(98)
[12] G. Pastor, M. Romera and F. Moic
Structure of One-Dimensional Quadratic Maps,” l
Review E, Vol. 56, 1997, pp. 1476-1483.
doi:10.1103/PhysRevE.56.1476
ntoya, “Harmon
Physica
[13] J. Hale and H. Koçak, “Dynd Bifurcations,” App- namics a
lied Mathematics, Vol. 3, 1991, p. 314.
doi:10.1007/978-1-4612-4426-4
[14] M. Misiurewicz and Z. Nitecki, “Combinatorial Patterns
for Maps of the Interval,” Memoirs of the American Ma-
thematical Society, Vol. 94, No. 456, 1991, pp. 109-110.
[15] A. Douady and J. H., “Etude Dynamique des
Polynômes Complexes,” Publications Mathematiques
Hubbard
1984 (Prem
ra, G. Pastor and F. Montoy a, “siurewicz P o in t s
Quadratic Maps,” A: Statis-
tical Mechanics and Its Applications, Vol. 232, No. 1-2,
i:10.1016/0378-4371(96)00127-6
d’Orsay, 84-02, ière Partie) and 85-04, 1985
(Deuxième Partie).
http://portail.mathdoc.fr/PMO/PDF/D_DOUADY_84_02.
pdf
http://mathdoc.emath.fr/PMO/PDF/D_DOUADY_85_04.
pdf
[16] M. RomeMi
in One-Dimensional Physica
1996, pp. 517-535. do
ra and F. Montoy
ewicz Patterns in One-D
A: Statist
8-4371(96)00128-8
[17] G. Pastor, M. Romea, “On the Calcula-
tion of Misiurimensional Quad-
ratic Maps,” Physica ical Mechanics and Its Ap-
plications, Vol. 232, No. 1-2, 1996, pp. 536-553.
doi:10.1016/037
[18] G. Pastor, M. Alvarez and F. Montoya, . Romera, G
292, No. 1-4, 2001,
pp. 207- 230. doi:10.1016/S0378-4371(00)00586-0
“Misiurewicz Point Patterns Generation in
One-Dimensional Quadratic Maps,” Physica A: Statistical
Mechanics and Its Applications, Vol.
[19] G. Pastor, M. Romera, J. C. Sanz-Martín and F. Montoya,
“Symbolic Sequences of One-Dimensional Quadratic
Maps Points,” Physica A: Statistical Mechs and Its
Applications, Vol. 256, No. 3-4, 1998, pp. 369-382.
doi:10.1016/S0378-4371(98)00083-1
anic
[20] A. Douady and J. H. Hubbard, “Itération ds Polynômes
Quadratiques Complexes,” C. R. Academic Society: Série
I, Paris, 1982.
Aca
with External Arguments in the Mandelbrot Set
Antenna,” Physica D: Nonlinear Phenome117,
No. 1-2, 2002, pp. 52-71.
doi:10.1016/S0167-2789(02)00539-0
e
[21] A. Douady, “Chaotic Dynamics and Fractals”demic
Press, New York, 1986.
[22] G. Pastor, M. Romera, G. Álvarez and F. Montoya, “Op-
erating
na, Vol.
n, “On Finite
nit Interval,”
Journal of Combinatorial Theory: Series A, Vol. 15, No.
1, 1973, pp. 25-44. doi:10.1016
[23] N. Metropolis, M. L. Stein and P. R. Stei
Limit Sets for Transformations on the U
/0097-3165(73)90033-2
[24] M. Morse and G. A. Hedl
ican Journal of Mathematics, Vol. 60, No. 4, 1938,
15-866. doi:10.2307/2371264
ung, “Symbolic Dynamics,”
Amer
pp. 8
[25] B.-L. Hao and W.-M. Zheng,
modal Maps Revisited,” Inte
, Vol. 3, No. 2, 1989, pp. 235-246.
42/S0217979289000178
“Symbolic Dynamics of Uni -
rnational Journal of Modern
Physics B
doi:10.11
der, “Fractals, Chaos, Power Laws,” W. H.
Freeman, New York, 1991.
Coexistence of Cycles of Continuous
Line into Itself,” Ukrainia
, 1964, pp. 61-71.
E. Yu. Roma-
adratic Maps,” Computer &
[26] M. Schroe
[27] A. N. Sharkovsky, “
Mapping of then Mathematical
Journal, Vol. 16
[28] A. N. Sharkovsky, Yu. L. Maistrenko and
nenko, “Difference Equations and Their Applications,”
Kluver Academic Publishers, Dordrecht, 1993.
[29] M. Romera, G. Pastor and F. Montoya, “Graphic Tools to
Analyse One-Dimensional Qu
Copyright © 2012 SciRes. IJMNTA
G. PASTOR ET AL.
Copyright © 2012 SciRes. IJMNTA
31
Graphics, Vol. 20, No. 2, 1996,39.
doi:10.1016/0097-8493(95)0013
pp. 333-3
4-4
nten
[30] M. Romera, G. Pastor and F. Montoya, “On the Cusp and
the Tip of a Midget in the Mandelbrot Set Ana,”
Physica Letters A, Vol. 221, No. 3, 1996, pp. 158-162.
[31] M. J. Feigenbaum, “Quantitative University for a Class of
Nonlinear Transformations,” Journal of Statistical Phys-
ics, Vol. 19, No. 1, 1978, pp. 25-52.
doi:10.1007/BF01020332
[32] R. L. Devaney, “Complex Dynamical Systems,” Ameri-
iden.
ring,” Internation
al of Bifurcation and Chaos, Vol. 13, No. 8, 2003,
pp. 2279-2300. doi:10.1142/S0218127403007941
can Mathematical Society, Provce, 1994
[33] M. Romera, G. Pastor, G. Alvarez and F. Montoya,
“Shrubs in the Mandelbrot Set Ordeal
Journ
toya, “Ex[34] G. Pastor, M. Romera, G. Alvarez and F. Mon-
ternal Arguments for the Chaotic Bands Calculation in the
Mandelbrot Set,” Physica A: Statistical Mechanics and Its
Applications, Vol. 353, No. 1, 2005, pp. 145-158
doi:10.1016/j.physa.2005.02.025 .
,
V. Fernández and F. Montoya, “of
Pseudoharmonics and Pseudoantiha
Mathematic and Computation, Vol. 213, No. 2, 2009, pp.
:10.1016/j.amc.2009.03.038
[35] G. Pastor, M. Romera, G. Alvare z, D. Arroy o, A. B. Orue
A General View
rmonics to Calculate
External Arguments of Douady and Hubbard,” Applied
484-497. doi
yo and F.
ce been Subshrubs and Chaotic
Bands in the Mandelbrot Set,” Disc a-
[36] G. Pastor, M. Romera, G. Alvarez, D. Arro
Montoya, “Equivalenetw
rete Dynamics in N
ture and Society, 2006, Article ID: 45920.
doi:10.1155/DDNS/2006/70471