Harmonic Analysis in Discrete Dynamical Systems

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 applications 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.


Introduction
In this paper we review a branch of harmonic analysis applied to discrete dynamical systems. In general, harmonic analysis has been widely used in experimental applications, as in the field of signal processing. In the same way, harmonic analysis applied to discrete dynamical systems can be a valuable tool for experimental scientists studying nonlinear phenomena. This paper is focused above all in showing some tools with interesting applications in nonlinear phenomena.
At its inception, the harmonic analysis studies the representation 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 applications 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- x c + = + . The logistic map [1][2][3] is widely known among experimental scientists studying nonlinear phenomena. Indeed, since Verhulst used it for the first time in 1845 to study population growth [1], it has served to model a large number of phenomena. The real Mandelbrot map is the intersection of the Mandelbrot set [4-6] and the real axis. All the 1D quadratic maps are topologically conjugate [7][8][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 representative paradigm of chaos. The Mandelbrot set can be defined as + , z and c complex, for the initial value 0 z = . 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][11][12] (likewise we could have used the logistic map), that can be defined again as  [13]. A similar definition of the multiplier λ can be given for the Mandelbrot 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 2 1 4 c − ≤ ≤ . In both cases, the real case and the complex one, when 1 λ ≠ one has hyperbolic points. The connected components of the c-values set for which ( ) 0 k c f converges to a k-cycle are periodic hyperbolic components (periodic HCs), or simply HCs [6]. These periodic HCs verify 1 λ < , which means they are stable (if 0 λ = 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 λ = ) 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 λ > which means they are unstable. These last points are, in addition, preperiodic and they have been later called Misiurewicz points [14][15][16][17][18]. When 1 λ = 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 cardioid-like component is born) and to pitchfork bifurcation points (or root points, where, in the case of a 2D quadratic 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 univocal. We are interested in names of HCs and Misiurewicz 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 CX … X (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 denominator 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, specifically 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 calculate the symbolic sequences of the period doubling cascade of the original orbit. Therefore, the MSS harmonics are a very valuable tool since, given the symbolic sequence 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 calculated. 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 antiharmonics (F antiharmonics), which in their subsequent papers 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 antiharmonics 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 sequences 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 orbits of the chaotic band generated by such an orbit.
As we have just said, in Sections 2.1 and 2.2 we introduce 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.
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 orderings and new calculations in this chaotic region.

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 calculate the symbolic sequences of the last appearance periodic 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.

MSS Harmonics
The symbolic dynamics is introduced by Morse and Hedlung 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 present paper is based on the paper of MSS and on the recipes of Schroeder [26].
The symbolic dynamics is based on the fact that sometimes it is not necessary to know the values of the iteration 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 CXX … X (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, x c + = + . 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 conjugate [7][8][9], therefore the logistic map and the real Mandelbrot 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, 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, 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 harmonics 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 harmonics 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 2 0 , 2 1 , 2 2 , 2 3 , 2 4 and 2 5 periodic orbits of the period doubling cascade, are: Note that, when i = 0, Let us see now the definition of antiharmonics, also introduced by MSS.
Let P be the pattern of a superstable orbit of the logistic map. The first MSS antiharmonic of P, ( ) ( )

MSS
A P , is formed by appending P to itself and changing the second C to L (or R) if the R-parity of P is even (or odd). The second MSS antiharmonic of P, ( ) ( ) 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-onical rule of a R-map. As in the case of the MSS harmonics, the periods of the successive MSS antiharmonics of a pattern P of period p are: 2p, 4p, 8p, … Antiharmonics 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. 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: that indeed do not correspond to any periodic orbit.

PRM Harmonics
In 1997 the PRM harmonics and PRM antiharmonics were introduced to contribute to the ordering of 1D quadratic 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]. Sharkovsky'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: where the symbol  must be 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]).
As said before, PRM harmonics were introduced in [12] to better understand the ordering and the generation of the symbolic sequences or patterns of superstable orbits. However, in the same way as the Sharkovsky ordering 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 superstable orbits. On the other hand, while MSS or Sharkovsky 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.
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 component (gene) and the corresponding chaotic band.
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-2 n chaotic band n B is the n th periodic orbit of the period doubling cascade, with period 2 n , which is the gene n G .  Let us consider the real Mandelbrot map, which as we 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 logistic 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, H P , is formed by appending P to itself and changing the second C to L (or R) if the L-parity of P is even (or odd). The second PRM harmonic of P, H P and changing the second C to L (or R) if the L-parity of ( ) ( ) 1 PRM H 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 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 harmonics 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 orbits. Let us note that, in the upper parts, symbolic sequences 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 C we add a C to the first one, i.e., we write CLC and we change the second C into a R, since the L-parity of CL is odd, and we obtain By applying the same procedure, we obtain that the third and fourth PRM harmonics of C are As is well known [12,26], CLR, CLR 2 , CLR 3 , … (CRL, CRL 2 , CRL 3 , … , in the logistic map) are the symbolic sequences of the last appearance superstable orbits. As it was already shown [12,30], the limit of the PRM harmonics of a pattern can be calculated. In this case the limit M , whose symbolic sequence is (CL)R (if C is taken into account, the preperiod is 2, and we have 2,1 M , as can be seen in [12]). This point is the tip (C) [12,30], whose parameter value is 2 c = − . As seen in the figure, we start from the period-1 superstable orbit C placed in the periodic region. The first PRM harmonic of C is the period-2 superstable orbit of the period doubling cascade. All the other PRM harmonics of C are superstable orbits placed in the period-2 0 chaotic band 0 B and are the last appearance superstable orbits of this band. If we consider the period-2 0 superstable orbit C as a gene 0 G , then the PRM harmonics of the gene 0 G generate the period-2 0 chaotic band 0 B . However, a period-2 1 superstable orbit placed in the periodic region is also generated. Let us see what happens when this orbit is used as a new gene 1 G . Let's now look at Figure 3 To form its PRM harmonics we add CL to the previous one and we change the second C into a L or a R, in accordance with the lero rule. So, we obtain that the first, second, third, and fourth PRM harmonics of 1 G are The limit of these harmonics is the Misiurewicz point ( ) [16,17], that separates the period-2 0 chaotic band 0 B and the period-2 1 chaotic band 1 B . The first harmonic generated from CL is the pattern CLRL of period-2 2 , which corresponds to the second superstable orbit of the period doubling cascade of the periodic region of C. All the other PRM harmonics of CL are superstable orbits placed in the period-2 1 chaotic band 1 B and, in addition, are the last appearance superstable orbits of this band. Again, if we consider CL as a gene 1 G , we have that the PRM harmonics of the gene 1 G generate the period-2 1 chaotic band 1 B (and a new gene 2 CLRL G = ). Let us see now Figure 3(c) where we show the PRM harmonics of the gene 2 CLRL G = . The first, second, third, and fourth harmonics of 2 G are the first being a new gene in the periodic region and the others superstable orbits placed in the period-2 2 chaotic band 2 B . The limit of these harmonics is the Misiurewicz [16,17] which separates the period-2 1 chaotic band 1 B and the period-2 2 chaotic band 2 B . Therefore the PRM harmonics of the gene 2 G generate the period-2 2 chaotic band 2 B , and a new gene, 3 3 CLRL RL G = , of the next chaotic band. Finally, let us see   G , and the last appearance superstable orbits of the period-2 3 chaotic band 3 B are generated. The limit of these superstable orbits is the Misiurewicz point   The patterns of the harmonic structure are called structural 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 infinity of chaotic bands B n , separated by Misiurewicz points called separators, m n , 0 n ≤ ≤ ∞ . Each structural pattern of each chaotic band and each separator is determined by starting with the only datum of the pattern C, and by applying the successive harmonics to the successive genes, what shows the power as calculation tool of the PRM harmonics.
As can be seen in detail in [12], if we deal with the logistic map, a R-map, instead of with the real Mandelbrot map, a L-map, the result is equivalent to what is shown in Figure 4. However, we have to take into account that in this case, the canonical direction of the logistic map is the rightward direction, the canonical parity is the R-parity and the canonical rule is the relo rule (therefore, we have to interchange Ls and Rs).
To finish this section, we shall see the PRM antiharmonics.
Let P be a pattern. The first PRM antiharmonic of P, A P is even (or odd). And so on.
As in the case of MSS antiharmonics, PRM antiharmonics are also a purely formal construction and never correspond to a periodic orbit either. But, although they have no real existence, they play an important role in 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 antiharmonics. If we were with the logistic map, a R-map, to obtain antiharmonics 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 the relo rule, and the canonical rule of a L-map is the lero rule. Harmonics (of MSS or PRM) are obtained by applying the canonical rules, and they grow in the canonical direction. Likewise, antiharmonics (of MSS or PRM) are obtained by applying the anticanonical rules, and they grow in the anticanonical direction.

Harmonics in 2D Quadratic Maps
In the same way as in Section 2 we reviewed harmonics (both MSS harmonics and PRM harmonics) in 1D quadratic maps, in this section we review both types of harmonics in 2D quadratic maps. The main difference between 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 Section 3.1 and PRM harmonics in Section 3.2.

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 2 1 ·p, 2 2 ·p, 2 3 ·p, ... In the same way as MSS introduced the concept of harmonics in 1D unimodal maps [23], by extension we call MSS harmonics of a HC of the Mandelbrot set to the set constituted by itself and all the discs of its period-doubling cascade (obviously we refer to their identifier).
Let ( ) 1 2 . , . a a 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 ( ) 1 2 . , . a a from the tuning algorithm of Douady and Hubbard [21,32]. Therefore, we can define the MSS harmonics as follows: . , . a a be the two EAs of a HC. The successive MSS harmonics of the HC are given by:  a a a a a a a a a a a a a a a   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 doubling cascade. Look at Figure 5 where .0100000111001110100000111010000100000111 as can be seen in Figure 6(b). Antiharmonics of MSS seem again to have no interest, as we can deduce next from their definition: . , . a a be the two EAs of a HC. The successsive MSS antiharmonics of the HC are given by:  a a a a a a a a  = , … Therefore, all of them are the same, which is the starting HC.

PRM Harmonics
Let us see now the PRM harmonics of a HC in this 2D case.

Figure 6. (a) Magnification of the branch 11 of the shrub (1/3) shown in Figures 5 (b) and (c), where the period-5 representative 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 ( ) 1 2 . , . a a be the EAs of a HC given in form of binary expansion [22]. The EAs of the order i PRM harmonics of ( ) 1 2 . , . a a are given by:  a a a a a a a a  a a  (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 =  , calculates a sequence of HCs which are the LAHCs of the corresponding chaotic band of the original HC. Indeed, let us analyze, as an example, the main cardioid that we consider as a gene 0 G (see Figure 7). By applying Equation (1) we obtain the successive PRM harmonics of 0 G . For 0 i = we obtain 0 G , a trivial case that normally not will be taken into account, for 1 i = we obtain a new gene 1 G in the periodic region, and for 2 i ≥ we obtain the LAHCs of the chaotic band 0 B . If we apply now Equation (2) to 0 G , we obtain the Misiurewicz point  − , which is the upper extreme of the chaotic band n B . That is, n G can be considered the gene or generator of the chaotic band n B . What we have just seen is similar to the harmonic 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 examples, the first one applied to the main cardioid (see Figure 7), and the second one to the period-5 midget of the branch 11 of the shrub (1 3 ) 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, 0 B , 1 B and 2 B , and also the main cardioid 0 G and the discs of the period doubling cascade, 1 G , 2 G , … . The EAs of the period-1 main cardioid, the gene 0 G , are By applying Equation (1)  m M = , which is the upper extreme of B 2 . And so on, such that, in a general case, G n generates the chaotic band B n (not shown in the figure).
Example 2 Let us see the period-5 midget of the branch 11 of the shrub(1 3 ) shown in Figure 6(b) which is a magnification of the rectangle b shown in Figure 6(a). Let us consider this period-5 cardioid as a gene whose EAs are If we apply Equation (1), we obtain the sequence Antiharmonics of PRM neither seem to have any interest, as in previous cases. Indeed, according to their definition: . , . a a be the EAs of a HC given in form of binary expansions [22]. The EAs of the order i PRM antiharmonics of ( ) 1 2 . , . a a are given by: The result in both equations is the starting HC. However, these concepts will be very useful in the next section.

Pseudoharmonics and Pseudoantiharmonics in the 2D Case
We have just seen that the PRM harmonics are a powerful 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.

Introduction of Pseudoharmonics and Pseudoantiharmonics
The introduction of pseudoharmonics and pseudoantiharmonics can be seen in detail in [35]. Let us first introduce pseudoharmonics. Let ( ) 1 2 . , . a a be the external arguments of a HC, and let ( ) . , . ; . , . . , .
. , . (8) calculates a pair of EAs of a Misiurewicz point, and not the starting HC as in the case of Equation (4). Again PRM antiharmonics are suitable to introduce pseudoantiharmonics that, as we shall see, are very useful to calculate more EAs.
Pseudoharmonics and pseudoantiharmonics are, as has been said, a generalization of PRM harmonics and PRM antiharmonics in the 2D case; therefore, these two last ones have to be a particular case of the two first ones. Indeed, when ( ) ( )

Some Considerations to Calculate Pseudoharmonics and Pseudoantiharmonics
To apply what we have seen so far, we shall briefly remember some concepts that can be seen in detail in [35].
We call a descendant [35] of ( ) to any of the HCs obtained from Equations (5) and (7). Likewise, the zone occupied by all the descendants calculated by using these equations is the zone of descendants [35]. In other to calculate a descendant, given the first pair, ( ) . , .
b b has to be an ancestor [35] of ( ) 1 2 . , . a a .
Let us briefly see the ancestors of ( ) . , .
c c itself is an ancestor of ( ) 1 2 . , . b b . Note that any HC is an ancestor of itself. Note also that the ancestor of a HC has lesser or equal period than the period of such a HC. In order to determine the ancestors of ( ) 1 2 . , . a a , next we shall remember some points.
As known from [33,36], the shrub of the n-ary hyperbolic component . , . a a be the representative of a branch (whose branch associated number is 1 2 m d d d  [33]) of i S .
The first ancestor of ( ) 1 2 . , . a a is the generator

Zones of Descendants
As can be seen in detail in [35], depending on the ancestor ( ) . , . . , . b b a a = .
As was already seen, when ( ) ( ) Equations (5), (6), (7) and (8) become Equations (1), (2), (3) and (4), hence pseudoharmonics and pseudoantiharmonics 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 associated to any cardioid, and alternatively the upper extreme and the LAHCs of the chaotic band associated to any disc.

Applications
Let us see an example for each of the four previous cases. Let us consider again the period-5 HC ( ) ( ) 1 2 . , . .00111, .01000 a a = , that is the representative of the branch 11 of the shrub (1 3 ). We help us with Figure  5, which shows such a shrub (1 3 ), and with .00111, .01000 in 1 S , that is the only subshrub of the chaotic region.

First Example
Let us begin with ( ) ( )

Third Example
Let us go on with ( ) ( )  M , which are the extremes of a sub-branch of the branch 11 (see Figure 6(a)). This sub-branch is called   . , . . , . .00111, .01000 a a b b = = (see Figure 6(b)). As always, This increasing part coincides with all the zone of descendants because the decreasing part of the zone of descendants becomes a point, the cusp of the original HC.

Conclusions
We review a branch of the harmonic analysis applied to dynamical systems. We focus on discrete dynamical systems, and more specifically on quadratic maps, both 1D quadratic maps and 2D quadratic maps, since these maps are the most widely used by experimental scientists, to whom we have addressed this paper. We first review the applications in 1D quadratic maps, in particular the harmonics and antiharmonics introduced by MSS. Given a periodic orbit, the MSS harmonics of the symbolic sequence of this orbit calculate the symbolic sequences of the period doubling cascade of the orbit. Metropolis, Stein and Stein introduced for the first time this type of calculation, and the concept of harmonics in dynamical systems.
Based on MSS harmonics, Pastor, Romera and Montoya introduced the PRM harmonics. These type of harmonics expand the possibilities of the calculations. 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 band where the original orbit is. Thus, if we start from the peiod-1 orbit, we can calculate the structure of a 1D quadratic map.
Both, MSS harmonics and PRM harmonics are redefined in order to can be used in 2D quadratic maps where external arguments are used as identifiers instead of symbolic sequences. Given the external arguments of a cardioid, the MSS harmonics calculate the EAs of the discs of its period doubling cascade, and the PRM harmonics the EAs of its LAHCs of its chaotic band 0 B . Pseudoharmonics and pseudoantiharmonics, which are an expansion of PRM harmonics and PRM antiharmonics in the case of 2D quadratic maps, are also reviewed. These new calculation tools are notably more powerful than PRM harmonics and PRM antiharmonics and make possi-ble the calculation of LAHCs in new zones of descendants.

Acknowledgments
This work was partially supported by Ministerio de Ciencia e Innovación (Spain) under grant TIN 2011-22668.