# Dynamics of Multimodal Families of m-Modal Maps

Authors: S. González-Salas; B. B. Cassal-Quiroga; J. Tuxpan; E. Campos
Journal: Complexity
 (2022)

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## Abstract

In this work, we introduce families of multimodal maps based on logistic map, i.e., families of m-modal maps are defined on an intervalI⊂ℝ, which is partitioned into non-uniform subdomains, with m∈ℕ. Because the subdomains of the partition are not uniform, each subdomain contains a unimodal map, given by the logistic map, that can have different heights. Therefore, we give the necessary and sufficient conditions for these modal maps present a multimodal family of m-modal maps, i.e., a bifurcation parameter can set a unimodal map, a bimodal map, up to a m-modal map. Some numerical examples are given according to the developed theory. Some numerical examples are given in accordance with the developed theory.


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## Body

## 1. Introduction

Many interesting results have been given in discrete dynamical systems, for example, in [1] the authors showed that period three implies whatever period so as a consequence chaos. Results about chaotic properties of non-autonomous discrete dynamical systems have been extensively studied in [2, 3]. Due to chaotic behavior presents ergodicity, sensitivity to initial conditions, transitivity, and not predictable evolution behaviors, chaotic dynamical systems have been considered with great potential in engineering applications. For example, to development pseudo-random number generators [4–6]; some applications of these generators are in video [7, 8], secure communications [9, 10], and cryptographic systems [11–15] due to the close relationship between chaos and cryptography. In [16, 17], a comparison between the properties of chaos and cryptosystems is given and showed that the ergodicity, sensitivity to initial conditions and the bifurcation parameter, deterministic dynamics and complex structure are analogous to confusion, diffusion, pseudo-randomness and algorithm complexity, respectively.Many proposals of bit generators have been developed with new discrete dynamical systems called maps which are capable of generating chaotic behavior via unidimensional systems, so there is a great interest in developing new chaotic maps for the purpose mentioned above or with the intention to understand the chaotic behavior [18]. Different maps have been used to tackle the aforementioned aims, like unimodal chaotic maps [19, 20], piecewise linear chaotic maps [21], and chaotic maps based on combining more that one chaotic map [10, 22]. To ensure the boundedness of chaotic trajectories, the systems are usually restricted to maps that are mapping from a compact interval into itself, usually the compact interval is I=0,1. However, there is no constraint to this interval, for example, in [23] the authors derived analytical expressions for the autocorrelation sequence and power spectral density of chaotic signals generated by one-dimensional continuous piecewise linear maps with three slopes f:−1,1⟶−1,1.Multimodal maps have been studied by Smania [24], who studied the dynamics of the renormalization operator for this kind of maps. Particularly, he developed a combinatory theory for a certain kind of multimodal maps. On the other hand, it is possible to generate multimodal maps based on unimodal maps like logistic map or tent map. Campos-Cantón et al. [25, 26] introduced multimodal maps based on the logistic map, but restricted to a regular partition of the space, i.e., the interval I is divided into uniform subintervals and each subinterval has a critical point that presents the same modal in all subintervals. The consequence of using a regular partition of the space is that the map shape in each interval is the same. In the same spirit that the previous work, mutimodal maps have been introduced based on the tent map [27], where analytical expressions have been derived for the autocorrelation function and the auto-spectral density function of chaotic signals generated by a multimodal skew tent map. In this work we present results on irregular partitions of space, this fact allows us to define different modals in all subintervals and obtain different maps. However, there are some map configurations that do not allow multimodal maps. So in this work we give the necessary and sufficient conditions for these modal maps present a multimodal family of m-modal maps, i.e., a bifurcation parameter can set a unimodal map, a bimodal map, up to a m-modal map.The useful insights for the study of chaos can be known by analyzing one-dimensional (1D) maps, e.g., the logistic and tent maps. These systems have been extensively studied [28] and implemented experimentally [29–34]. A unimodal map is a continuous 1D function ℝ⟶ℝ with a single critical point c0, monotonically increasing on one side of c0 and decreasing on the other. Here, we introduce a class of multimodal maps which is obtained by translating and scaling the logistic maps based on the bi-parametric equation proposed by Verhulst in 1838 [35].(1)dNdt=αg1−NγcN,where Nt is the state of the system at time t, αg is the intrinsic growth rate, and γc is the carrying capacity.Motivate by a large number of applications of the chaotic maps, we study a multimodal maps family based on the logistic map. The importance of having new unidimentional chaotic maps helps to understand the properties of chaotic behavior in discrete dynamical systems and allows applications such as new proposal of bit generators. Now, in this work we study multimodal maps considering irregular partitions of the space. Also the critical points present different local modals giving extra degrees of freedom for sequences generation. The development of this new class of families of chaotic map are based on piecewise continuous function. The map is defined such that only one parameter is considered to takes different values, the others are fixed, thus we will be working with monoparametric families formed with multiple unimodal mappings.The article is organized as follows: In Section2, we set the basis of m-modal maps and give its definition. In Section 3, we define a multimodal family of m-modal maps and give the necessary and sufficient conditions for these families to behave as a unimodal map, a bimodal map, up to a m modal map. In Section 4, the conditions under a multimodal family of maps can show transitivity are given and the analysis of the fixed points is presented, as well. In Section 5, numerical examples are given through a trimodal map. Finally, in the last Section 6 conclusions are given.


## 2. m-Modal Maps

LetI,d be the compact metric space I=0,1 endowed with the Euclidean metric. It is given the next definition:Definition 1.
LetS and f be an interval and a function, respectively, such that S⊂I and f∈CS,I. If there exists an xc∈S such that for all x∈S we have that f|x<xc is strictly increasing and f|xc<x is strictly decreasing then f is called a unimodal map on S.
Logistic map and tent map are examples of unimodal mapsI=S. A bimodal map can be defined by considering I=S1∪S2, S1∩S2=∅, such that the map f is a unimodal map in each interval S1⊂I and S2⊂I. Therefore, there is a map with two modals on I, called a bimodal map on I. In this way, it is possible to generalize to an arbitrary number of modals to get a m-modal map based on a unimodal map. The interest of this work is to build piecewise functions to generate m-modal maps based on a unimodal map. Particularly in this work, the logistic map is used, which is given as follows:(2)fγx=γβ−xx−α,withx∈S=α,β⊂I,where x is the state variable; γ∈ℝ is a bifurcation parameter, and α, β are fixed arbitrary parameters with restricted values 0≤α<β≤1. The first and second derivatives of fγ are fγ′=γ−2x+α+β and fγ″=−2γ, respectively, then fγ is continuous for all x∈ℝ and has a local maximum or a local minimum at xc=α+β/2, fγ′xc=0, for γ≠0. Therefore, to get a unimodal map on the interval α,β based on the logistic map, it is necessary to consider γ>0γ∈ℝ+.
In the case of m-modal maps based on a unimodal map, the intervalI is divided into multiple subintervals, where each subinterval contains its own maximum, and every subinterval can have the same or different length. Let us consider a partition of the interval I as follows:Definition 2.
LetΠ be a partition of I which is determined by a finite sequence ζii=0m, with m∈ℕ\1,such that ζ0=0<ζ1<⋯<ζm=1:(3)Π=S1=[ζ0,ζ1],S2=(ζ1,ζ2],…,Sm−1=(ζm−2,ζm−1],Sm=(ζm−1,ζm].
If there exists at least a pair of subintervalsSi and Sj, with i≠j, such that σi≠σj, where σi=dζi−1,ζi is the diameter of Si, with i,j∈1,2,…,m, then the partition on I is called nonuniform. In the contrary case, the partition on I is called uniform.
To illustrate the above definition, it is given the following example.Example 1.
Consider the following sequence of points onI,
ζii=03=0,1/2,2/3,1, we have ζ0=0<ζ1=1/2<ζ2=2/3<ζ3=1 which determine the partition of I, Π=0,1/2],(1/2,2/3],(2/3,1, where σ1=1/2≠σ2=1/6≠σ3=1/3, therefore, the partition is nonuniform.
Now, a m-modal map is defined based onm unimodal maps as follows.Definition 3.
Letg∈CI,R and Π=S1,…,Sm be a map and a partition on I with m∈ℕ1,respectively. If the map g is unimodal on each Si∈Π, with i=1,…,m, then it is called m-modal map.
Now, we construct a m-modal map based on the unimodal maps given by Equation (2), which are defined on all subintervals Si of the partition Π of I. To determine the piecewise function that defines the m-modal map, we start by giving the sequence of points ζii=0m=ζ0,ζ1,ζ2,…,ζm, which determine a partition Π on I. Therefore, the continuous piecewise function is defined as follows:(4)hγx=γζ1−xx−ζ0,forζ0≤x≤ζ1;ζ2−xx−ζ1,forζ1<x≤ζ2;⋮⋮ζm−xx−ζm−1,forζm−1<x≤ζm..
Equation (4) defines a map with m modals, where γ is the bifurcation parameter. Recall that the modal of each subinterval Si is given by xci=ζi−1+ζi/2, for i=1,…,m. Then, the local maximum of the map (4) is given by hxci=γ/4σi2,in each Si. In the case that the partition Π on I is uniform, each σi=1/m with i∈1,2,…,m and the local maximum of each unimodal map in Equation (4) is the same given by γ/4m2.But if Π is nonuniform, then there exists at least a pair of subintervals Si with different diameter σi≠σj; therefore, for a given value of γ, the m-modal map (4) presents multiple local maximums hxci≠hxcj. Notice that the local maximums are determined by the parameter γ and the diameter σi of each subinterval Si. The interest is to control the local maximums independently of each σi, so a new parameter ρi is considered in Equation (4). To avoid the effect of the parameter σi to determine the local maximum, then we consider ρi/σi2 in each piece of the m-modal map (4). A m-modal map is given as follows:(5)gγx=γρ1σ12ζ1−xx−ζ0,forζ0≤x≤ζ1;ρ2σ22ζ2−xx−ζ1,forζ1<x≤ζ2;⋮⋮ρmσm2ζm−xx−ζm−1,forζm−1<x≤ζm;,where ρi>0, with i=1,…,m, are arbitrary fixed parameters but restricted to gγ:∪i=1kSi⟶∪i=1kSi, for 1≤k≤m, for all γ∈(0,4].Lemma 1.
Maximumρ-value: Consider a m-modal map given by (5). If g4:I⟶I, then at least one ρi=1 and the others ρi∈(0,1], with i=1,…,m.Proof.
Forγ=4, the local maximums are given by ρ1,ρ2,…,ρm. Because g4:I⟶I then at least one ρi=1 and none of them is greater than 1. If the ρ’s are different ρ1≠ρ2≠⋯≠ρm, then only one ρi=1 and the others ρi∈0,1, with i=1,…,m. If all ρ’s are not different then this allows the possibility of having more ρ′s=1. However, they can be always ordered as ρj1≤ρj2≤⋯≤ρjm−1≤ρjm, with j1,j2,…,jm=1,2,…,m. This configuration allows to have at least one ρi=1, and the others ρi∈(0,1]. The proof is completed.


## 3. Multimodal Family of m-Modal Maps

One of the main contributions of this work is the definition of the multimodal family of m-modal maps. However, we are also interested in those families of m-modal maps that are able to behave like a unimodal map, a bimodal map, etc., up to a m-modal map according to the bifurcation parameterγ. In other words, these monoparametric families gγ behave like a k-modal map, with k=1,…,m, if it is possible to choose the appropriate values γ1,γ2,…,γm for the bifurcation parameter γ which makes gγk behaves like a unimodal map, a bimodal map, up to a m-modal map, respectively. The property is that ∪i=1kSi⊂I will be invariant under each gγk such that arbitrary length orbits could be calculated from iterating any initial value x0∈I, i.e., for each γk, the interval 0,ζk would be invariant under gγk.Definition 4.
Letgγk∈CI,R be a continuous m-modal map given by (5), where m∈N\1,if there exist m different values for the bifurcation parameter γ∈γ1,γ2,…,γm−1,γm such that they fulfill(i)
gγk:0,ζk⟶0,ζk is surjective, with k=1,…,m.
The monoparametric familygγkk=1m is called multimodal family of m-modal maps.
The above definition allows us to see that to obtain a multimodal family, parameter values ofζii=0m, and ρii=1m should determine the existence of parameters γk. So, in the next theorem, necessary and sufficient conditions are given to construct multimodal families by using m-modal maps. Considering that for γ=4, gγ is surjective on I.Theorem 1. (configurations).
Letgγx be a m-modal map given by equation (5) on I with γ=4. There exist control parameter values γ1,γ2,…,γm such that the family gγkk=1m is a multimodal family of m-modal map if and only if:(6)ζ1ρ1≠ζ2maxi≤2ρi≠⋯≠ζm−1maxi≤m−1ρi,and for every j∈1,2,…,m−1:(7)ζjmaxi≤jρi<1.Proof.
⇐ We have that g4 is a m-modal map and is sujerctive on I and suppose that conditions given by equations (6) and (7) are fulfilled; it must be proved that there exist m different control parameter values γ1,γ2,…,γm, such that gγi:0,ζi⟶0,ζi are surjective, i.e., gγiI⊆I, where i∈1,2,…,m .
Asg4 is a m-modal map function given by (5), then each subinterval of the partition Π=S1,S2,…,Sm on I satisfies that g4ζi=0, g4xci=ρi and ρi>0, due to they are local maximums. According to Corollary 1, as g4 is surjective on I there exists at least a local maximum ρi=1 and it is also satisfied that every ρi≤1.
The local maximum undergγ on each subinterval of Π is given by(8)gγxci=ρiγ4.
Thus, whengγ is restricted to the interval Jk=0,ζk, then there are k local maximums ρ1γ/4 , ρ2γ/4, ⋯, ρkγ/4, with k=1,…,m. Since for each local maximum, the parameter value of γ is the same, i.e., it is a constant, then the maximum value of gγ restricted to the interval Jk=0,ζk is(9)maxx∈Jkgγx=γ4maxj≤kρj.
We are finding a parameter valueγk∈0,4 such that maxx∈Jkgγkx=ζk. Doing some algebra, it results that(10)γk=4ζkmaxj≤kρj.
Due to (7),γk<4, for k∈1,2,…,m−1, and the local maximum of gγk on Jk=0,ζk is(11)maxx∈Jkgγkx=4maxj≤kρjmaxj≤kρj4ζk=ζk.
Asgγk is continuous on Jk then it is satisfied gγk is surjective on Jk=0,ζk; besides, as k is arbitrary then there exist m control parameters γk such that gγk:0,ζk⟶0,ζk are surjective, with k=1,…,m. By equations (10) and (6) we have:(12)γ1=4ζ1ρ1≠γ2=4ζ2maxj≤2ρj≠⋯≠γm−1=4ζm−1maxj≤m−1ρj≠γm=4,so γi≠γj for all i≠j. Then there are m different values for the bifurcation parameter γ: γ1≠γ2≠⋯≠γm−1≠γm and I is invariant under each gγk where k=1,2,…,m; therefore the family gγk is a multimodal family of a m-modal maps.⇒
Now, we have thatgγk=1m, for γ∈(0,4], is a multimodal family of m-modal maps given by (5), so g4 is surjective on I and there exists γk such that gγk:0,ζk⟶0,ζk, for k=1,…,m. We need to prove that the existence of γk implies that equations (6) and (7) are fulfilled.
We know that for eachγk there are k local maximums at gγkxc1=ρ1γk/4, gγkxc2=ρ2γk/4, ⋯, gγkxck=ρkγk/4. Because gγk is surjective on Jk, then it is satisfied that maxx∈Jigγkx=ζk. As a consequence ρiγk/4≤ζki∈1,2,…,k and maxj≤kρjγk/4=ζk. So the control parameter value is given by γk=4ζk/maxj≤kρj, then γk/4=ζk/maxj≤kρj. We also know that there are m different values of γ: γ1≠γ2≠…≠γm−1≠γm. Since dividing the γ values by four does not affect the inequalities, then the condition (6) is fulfilled.(13)ζimaxl≤iρl≠ζjmaxl≤jρl.
Due toγk∈0,4, for k=1,…,m−1 and γm=4, then γk<γm, for k=1,…,m−1. So γk/4<γm/4=1, therefore(14)ζjmaxi≤jρi<1,
which fulfills equation (7) and the proof is completed.Example 2.
As an example, it is selectedζii=03=0,0.2,0.6,1 and ρii=13=1,0.7,0.6 which determine a nonuniform partition of I and the local maximum for each subinterval Sii=13. With the aforementioned parameters, a trimodal map is defined by (5) as follows:(15)gγkx=γk250.2−xx−0,for0≤x≤0.2;4.3750.6−xx−0.2,for0.2<x≤0.6;3.751−xx−0.6,for0.6<x≤1;,
becausemaxj≤3ρi=1, then the trimodal map g4 is surjective on I. Besides, it is easy to check that the trimodal map given by (15) fulfills the equation (6) and (7) of Theorem 1; thus there exist control parameters that satisfy Definition 4 by calculating them with (10) results in γ1=0.8, γ2=2.4, and γ3=4. Therefore, gγ given by (15) is a multimodal family comprises by a unimodal map, bimodal map and trimodal map, i.e., g0.8:0,0.2⟶0,0.2 defines a unimodal map, g2.4:0,0.6⟶0,0.6 defines a bimodal map, and g4:I⟶I defines a trimodal map. It is worth mentioning that the trimodal map does not have a chaotic behavior for all the considered values of γ’s. The Lyapunov exponents are 0.6931, -1.0312, and 1.1162 for γ1=0.8, γ2=2.4, and γ3=4, respectively. Note that for γ2=2.4 the Lyapunov exponent is -1.0312, then the orbits converge at a fixed point. The interest is to consider multimodal families with chaotic behavior for all considered values of γ’s. Therefore, we need to consider values of γ such that the multimodal families present unstable fixed points.Corollary 1. (uniform partition).
Letgγx be a m-modal map given by equation (5) on I with γ=4, uniform partition Π, and ρ1=ρ2=⋯=ρm. Then there exist control parameter values γ1,γ2,…,γm such that the family gγkk=1m is a multimodal family of m m-modal map.Proof.
This corollary is a direct consequence of Theorem1.


## 4. Dynamics of the Multimodal Family of m-Modal Maps

Once the necessary and sufficient conditions to choose a multimodal family of m-modal maps are given, the next step is to analyze the dynamics of each m-modal map of a family. As the interest of this work is to develop chaotic behavior, the m-modal maps must display chaotic behavior in all of its members of the family. In this section, the necessary and sufficient conditions are given to avoid only regular motion and obtain a multimodal family of chaotic maps.


### 4.1. Fixed Points Analysis

It is well known that there exist orbits of maps which dynamics are not useful to create chaotic behaviour, so it is better to avoid them. For example, when a map convergences asymptotically to a stable fixed point, it generates a time series without fluctuations, useless to produce chaotic behavior.To avoid the existence of multimodal families with stable fixed points, it is made a local stability analysis and the result is given in the following theorem.Theorem 2. (stable fixed points).
Letgγx be a m-modal map given by equation (5) on I if the following interval exists(16)Υj=ζj−1+ζj+2ζj−1ζjρj,ζj−1+ζj+2σj2+ζj−1ζjρj,and γj∈Υj, then two fixed points of system (5) exist in each subinterval ζj−1,ζj and at least one is stable.Proof.
We start by calculating the fixed points of a multimodal family withm maps gγjj=1m, then the following equation is solved(17)xn+1=gγjxn=xn,forxn∈I,where gγk is a multimodal map given by (5) and defined in m intervals S1=ζ0,ζ1, S2=(ζ1,ζ2], …, Sm=(ζm−1,ζm]. The fixed points are calculated by the following equation(18)γjρjσj2ζj−xnxn−ζj−1=xn,forxn∈Sj,where j∈1,2,…,m. Each part of the function must have a maximum of two fixed points; thus an m-modal map must have a maximum of 2m fixed points. The fixed points are given as follows:(19)x¯L,Rj=σj2−γjρjζj+ζj−1±σjγj2ρj2−2γjρjζj+ζj−1+σj2−2γjρj.
There are two fixed points if the discriminant is positive.γk>0 for each mutimodal family, thus the fixed points exist if the following inequality is fulfills:(20)γk>ζj−1+ζj+2ζj−1ζjρj.
Now, we need to prove that at least one of the fixed points is stable. In our casex¯Rj is stable if gγj′x¯Rj<1 for gγ defined in Sj which is satisfied when(21)γk∈ζj−1+ζj+2ζj−1ζjρj,ζj−1+ζj+2σj2+ζj−1ζjρj.
The proof is completed.
The inequality (20) warranties the existence of two fixed points in each interval Si, with i=1,…,m. However, when gγj domain is restricted to the interval S1=ζ0=0,ζ1, there always exists the fixed point x¯L1=0 which is asymptotically stable when the condition gγ1′x¯L1<1 is fulfilled.(22)gγ1′0=−2γkρ1σ120+γkρ1σ12ζ0+ζ1=γkρ1ζ1<1.
This results inγ1∈0,ζ1/ρ1, then x¯L1=0 is asymptotically stable and is unique. When γk>ζ1/ρ1 there exist two fixed points and the second fixed point is given by(23)x¯R1=γ1ζ1ρ1−ζ12γ1ρ1.
This fixed pointx¯R1 is asymptotically stable always that γ1∈ζ1/ρ1,3ζ1/ρ1 is fulfilled, according to (21).Definition 5.
The setΥ is given by the union of the intervals 0,ζ1/ρ1 and Υj, if Υj exists. Thus, Υ is called the stable set of γ parameter values.
Therefore, ifγ∈Υ, then the m-modal map gγx given by (5) on I presents at least one stable fixed point.Definition 6.
The setΥ∗=0,4−Υ¯ is called the set of γ parameter values such that the m-modal map gγx given by Equation (5) on I presents unstable fixed points, where Υ¯ is the closure of Υ.
If anyγk of the multimodal family gγ1,gγ2,…,gγm belongs to the set Υ, then the data series from the multimodal map are useless to generate chaotic behaviour.


### 4.2. Transitivity

One of the important characteristics of dynamical systems is the transitivity property, which describes that given any open subsetsU1,U2∈X, there exists an x0∈U1, and n>0, such that fnx0∈U2. It is worth to mention that Theorem 1 only states the parameters useful to get a multimodal family of maps, but these maps could be non-transitive. Now, it is important to establish the necessary conditions to obtain the transitivity property for each map of a multimodal family gγkk=1m on the interval J=0,ζk, after the parameter values ζi and ρi are selected, so the transitivity is warrantied by the next proposition:Theorem 3.
Letgγkk=1m be a multimodal family of m-modal maps given by equation (5) on I, with γk∈Υ∗. Then the multimodal family of m-modal maps is transitive for γ1,…,γm. If any of the following cases occur:
(a)(24)gγkxc1=gγkxc2=⋯=gγkxcm,withk=1,…,m.
(b)(25)gγkxc1>gγkxc2>…>gγkxcm,withk=1,…,m.and x¯Lj>gγk2xcj for all j=2,…,m, if there exist x¯Lj and gγkxcj in the Sj interval.(c)(26)gγkxc1<gγkxc2<…<gγkxcm,withk=1,…,m.and the following inequality is always preserved(27)gγkxcj>xLj+1,forj=1,…,m−1,where xcj and x¯Lj are the critical point and the left fixed point in the j-th interval, with j=1,…,m, and x¯Lj+1 is the fixed point in the Sj+1 interval.Proof.
See appendix A.
If the m-modal map presents combined modals, i.e.,
gγkxcj>gγkxcj+1<gγkxcj+2, with j=1,…,m−2, then Theorem 3 (b) and (c) needs to be checked.Example 3.
Analysis of an example of transitivity of multimodal maps. Specifically, the aim is to design a trimodal family of transitive mapsgγii=13, such that Theorem 3 (c) is fulfilled. Therefore, the parameter m=3 determines that the interval I is partitioned in three subintervals. Arbitrarily, we propose the following parameters to generate the partition: ζii=03=0,1/4,1/2,1. We also know that gγ3xc1<gγ3xc2<gγ3xc3=1, then ρ3=1 and ρ3/σ32=4. Therefore, gγ3xn is defined in the subinterval 0.5,1 as follows:(28)xn+1=γk⋅41−xnxn−0.50,the fixed points x¯L3 and x¯R3 are given by 0.5899 and 0.8475, respectively. Accordingly Theorem 3 c)0.5899<ρ2 < 1, so we set ρ2=0.7 that generates ρ2/σ22=11.2. Now gγ3xn is defined in the subinterval 0.25,0.5 as follows:(29)xn+1=γk⋅11.20.5−xnxn−0.25,the fixed points x¯L2 and x¯R2 are given by 0.2779 and 0.4497, respectively. Then 0.2779<ρ1 < 0.7, so ρ1=0.5 and ρ1/σ22=8. Therefore, gγ3xn is defined in the subinterval 0,0.25 as follows:(30)xn+1=γk⋅80.25−xnxn−0,the fixed points x¯L2 and x¯R2 are given by 0 and 0.2187, respectively. Therefore, ρii=13=0.5,0.7,1 forms a trimodal map given as follows:(31)xn+1=gγkxn=γk80.25−xnxn,for0≤xn≤0.25;11.20.50−xnxn−0.25,for0.25<xn≤0.50;41−xnxn−0.50,for0.50<xn≤1..
Due tomaxi≤3ρi=1, ζ1/ρ1=0.5<ζ2/maxj≤2ρi=0.7143<1, conditions of Theorem 1 are fulfilled, then there exist control parameter values that form a multimodal family which are γ1=2, γ2=2.8571 and γ3=4, see Figure 1. Meanwhile fixed points and maximum values of this family of multimodal maps g2, g2.8571, and g4 are shown in Table 1.The cobweb diagrams of the maps: gγ3=4:I⟶I, see Figure 1(a); gγ2=2.8571:0,0.5⟶0,0.5, see Figure 1(b); and gγ1=2:0,0.25⟶0,0.25 (see Figure 1(c)), using the (31) with x0=0.05. These maps gγi are surjective on Ji=0,ζi, with i=1,2,3; besides, these maps fulfill theorem (3). Then maps gγ1, gγ2 and gγ3 are transitive on 0,ζ1, 0,ζ2 and I, respectively, as is shown in Figure 1. Figure 2 shows the bifurcation diagram where it is possible to observe that period 2 is a route to chaos because the family is based on the logistic map.
The Lyapunov exponents of these sequences are 0.6932, 1.0127 and 1.3374 forγ1=2, γ2=2.85, and γ3=4, respectively. Then, the family composed of a unimodal map, a bimodal map, and a trimodal map is given by (31) exhibits chaotic behavior.Figure 1
Cobweb diagram of the maps: (a)gγ3=4:I⟶I, (b)gγ2=2.8571:0,0.5⟶0,0.5 and (c)gγ1=2:0,0.25⟶0,0.25, using equation (31) with x0=0.05 which a long evolution covers completely the intervals I, 0,ζ2 and 0,ζ1, for γ3, γ2, γ1, respectively.
(a)(b)(c)Table 1
Maximum values and fixed points of the multimodal familygγkk=13, for γ1=2, γ2=2.85, and γ3=4.
kgγkxc1gγkxc2gγkxc3x¯L,R1x¯L,R2x¯L,R310.250.350.50, 0.1875−,−−,−20.35630.50.71430, 0.20620.2949, 0.4238−,−30.50.710, 0.21870.2779, 0.44970.5899, 0.8475Figure 2
Bifurcation diagram of system (31).Example 4.
Analysis of a non-transitivity example, we define a multimodal family with the following parameters:m=3, ζii=03=0,1/4,1/2,1. ρii=13=0.26,0.51,1, which form a trimodal map given as follows:(32)xn+1=gγkxn=γk4.160.25−xnxn,for0≤xn≤0.25;8.160.50−xnxn−0.25,for0.25<xn≤0.50;41−xnxn−0.50,for0.50<xn≤1..
Notice that the values ofρii=13 do not fulfill Theorem 3 (c), only the conditions of Theorem 1 are fulfilled, maxi≤3ρi=1, ζ1/ρ1=0.9615<ζ2/maxj≤2ρi=0.9804<1. Then there exist control parameter values that form a multimodal family which are γ1=3.8462, γ2=3.9216 and γ3=4, see Figure 3(a)–3(c)), respectively. Meanwhile, fixed points and maximum values of this family of multimodal familygγii=13 are shown in Table 2.
Each mapgγi is surjective on Ji=0,ζi, with i=1,2,3; however, all of them are not transitive because Theorem 3 is not fulfilled. Then the maps gγ2, gγ3 are not transitive on 0,ζ2 and I, respectively. Only the map gγ1 is transitive, see Figure 3(c).
Figure4 shows the bifurcation diagram of system (32), where it is possible to observe that this bifurcation diagram resembles the bifurcation diagram of the logistic map. It is worth mentioning that the initial condition used to compute the bifurcation diagram is x0=0.3. Multistability was numerically observed for this system, however, multistability is not addressed in this work because it is off target.Figure 3
Cobweb diagrams of the mapsgγii=13 given by equation (32) with x0=0.9 for: (a)γ3=4, (b)γ2=3.9216 and (c)γ1=3.8462.
(a)(b)(c)Table 2
Maximum values and fixed points of the multimodal familygγkk=13, for γ1=3.846, γ2=3.922, and γ3=4.
kgγkxc1gγkxc2gγkxc3x¯L,R1x¯L,R2x¯L,R310.250.490.9620, 0.1880.296, 0.4220.596, 0.83920.2550.50.980, 0.1890.295, 0.4240.593, 0.84330.260.5110, 0.190.294, 0.4260.59, 0.848Figure 4
Bifurcation diagram of system (32) by using x0=0.3.


## 4.1. Fixed Points Analysis

It is well known that there exist orbits of maps which dynamics are not useful to create chaotic behaviour, so it is better to avoid them. For example, when a map convergences asymptotically to a stable fixed point, it generates a time series without fluctuations, useless to produce chaotic behavior.To avoid the existence of multimodal families with stable fixed points, it is made a local stability analysis and the result is given in the following theorem.Theorem 2. (stable fixed points).
Letgγx be a m-modal map given by equation (5) on I if the following interval exists(16)Υj=ζj−1+ζj+2ζj−1ζjρj,ζj−1+ζj+2σj2+ζj−1ζjρj,and γj∈Υj, then two fixed points of system (5) exist in each subinterval ζj−1,ζj and at least one is stable.Proof.
We start by calculating the fixed points of a multimodal family withm maps gγjj=1m, then the following equation is solved(17)xn+1=gγjxn=xn,forxn∈I,where gγk is a multimodal map given by (5) and defined in m intervals S1=ζ0,ζ1, S2=(ζ1,ζ2], …, Sm=(ζm−1,ζm]. The fixed points are calculated by the following equation(18)γjρjσj2ζj−xnxn−ζj−1=xn,forxn∈Sj,where j∈1,2,…,m. Each part of the function must have a maximum of two fixed points; thus an m-modal map must have a maximum of 2m fixed points. The fixed points are given as follows:(19)x¯L,Rj=σj2−γjρjζj+ζj−1±σjγj2ρj2−2γjρjζj+ζj−1+σj2−2γjρj.
There are two fixed points if the discriminant is positive.γk>0 for each mutimodal family, thus the fixed points exist if the following inequality is fulfills:(20)γk>ζj−1+ζj+2ζj−1ζjρj.
Now, we need to prove that at least one of the fixed points is stable. In our casex¯Rj is stable if gγj′x¯Rj<1 for gγ defined in Sj which is satisfied when(21)γk∈ζj−1+ζj+2ζj−1ζjρj,ζj−1+ζj+2σj2+ζj−1ζjρj.
The proof is completed.
The inequality (20) warranties the existence of two fixed points in each interval Si, with i=1,…,m. However, when gγj domain is restricted to the interval S1=ζ0=0,ζ1, there always exists the fixed point x¯L1=0 which is asymptotically stable when the condition gγ1′x¯L1<1 is fulfilled.(22)gγ1′0=−2γkρ1σ120+γkρ1σ12ζ0+ζ1=γkρ1ζ1<1.
This results inγ1∈0,ζ1/ρ1, then x¯L1=0 is asymptotically stable and is unique. When γk>ζ1/ρ1 there exist two fixed points and the second fixed point is given by(23)x¯R1=γ1ζ1ρ1−ζ12γ1ρ1.
This fixed pointx¯R1 is asymptotically stable always that γ1∈ζ1/ρ1,3ζ1/ρ1 is fulfilled, according to (21).Definition 5.
The setΥ is given by the union of the intervals 0,ζ1/ρ1 and Υj, if Υj exists. Thus, Υ is called the stable set of γ parameter values.
Therefore, ifγ∈Υ, then the m-modal map gγx given by (5) on I presents at least one stable fixed point.Definition 6.
The setΥ∗=0,4−Υ¯ is called the set of γ parameter values such that the m-modal map gγx given by Equation (5) on I presents unstable fixed points, where Υ¯ is the closure of Υ.
If anyγk of the multimodal family gγ1,gγ2,…,gγm belongs to the set Υ, then the data series from the multimodal map are useless to generate chaotic behaviour.


## 4.2. Transitivity

One of the important characteristics of dynamical systems is the transitivity property, which describes that given any open subsetsU1,U2∈X, there exists an x0∈U1, and n>0, such that fnx0∈U2. It is worth to mention that Theorem 1 only states the parameters useful to get a multimodal family of maps, but these maps could be non-transitive. Now, it is important to establish the necessary conditions to obtain the transitivity property for each map of a multimodal family gγkk=1m on the interval J=0,ζk, after the parameter values ζi and ρi are selected, so the transitivity is warrantied by the next proposition:Theorem 3.
Letgγkk=1m be a multimodal family of m-modal maps given by equation (5) on I, with γk∈Υ∗. Then the multimodal family of m-modal maps is transitive for γ1,…,γm. If any of the following cases occur:
(a)(24)gγkxc1=gγkxc2=⋯=gγkxcm,withk=1,…,m.
(b)(25)gγkxc1>gγkxc2>…>gγkxcm,withk=1,…,m.and x¯Lj>gγk2xcj for all j=2,…,m, if there exist x¯Lj and gγkxcj in the Sj interval.(c)(26)gγkxc1<gγkxc2<…<gγkxcm,withk=1,…,m.and the following inequality is always preserved(27)gγkxcj>xLj+1,forj=1,…,m−1,where xcj and x¯Lj are the critical point and the left fixed point in the j-th interval, with j=1,…,m, and x¯Lj+1 is the fixed point in the Sj+1 interval.Proof.
See appendix A.
If the m-modal map presents combined modals, i.e.,
gγkxcj>gγkxcj+1<gγkxcj+2, with j=1,…,m−2, then Theorem 3 (b) and (c) needs to be checked.Example 3.
Analysis of an example of transitivity of multimodal maps. Specifically, the aim is to design a trimodal family of transitive mapsgγii=13, such that Theorem 3 (c) is fulfilled. Therefore, the parameter m=3 determines that the interval I is partitioned in three subintervals. Arbitrarily, we propose the following parameters to generate the partition: ζii=03=0,1/4,1/2,1. We also know that gγ3xc1<gγ3xc2<gγ3xc3=1, then ρ3=1 and ρ3/σ32=4. Therefore, gγ3xn is defined in the subinterval 0.5,1 as follows:(28)xn+1=γk⋅41−xnxn−0.50,the fixed points x¯L3 and x¯R3 are given by 0.5899 and 0.8475, respectively. Accordingly Theorem 3 c)0.5899<ρ2 < 1, so we set ρ2=0.7 that generates ρ2/σ22=11.2. Now gγ3xn is defined in the subinterval 0.25,0.5 as follows:(29)xn+1=γk⋅11.20.5−xnxn−0.25,the fixed points x¯L2 and x¯R2 are given by 0.2779 and 0.4497, respectively. Then 0.2779<ρ1 < 0.7, so ρ1=0.5 and ρ1/σ22=8. Therefore, gγ3xn is defined in the subinterval 0,0.25 as follows:(30)xn+1=γk⋅80.25−xnxn−0,the fixed points x¯L2 and x¯R2 are given by 0 and 0.2187, respectively. Therefore, ρii=13=0.5,0.7,1 forms a trimodal map given as follows:(31)xn+1=gγkxn=γk80.25−xnxn,for0≤xn≤0.25;11.20.50−xnxn−0.25,for0.25<xn≤0.50;41−xnxn−0.50,for0.50<xn≤1..
Due tomaxi≤3ρi=1, ζ1/ρ1=0.5<ζ2/maxj≤2ρi=0.7143<1, conditions of Theorem 1 are fulfilled, then there exist control parameter values that form a multimodal family which are γ1=2, γ2=2.8571 and γ3=4, see Figure 1. Meanwhile fixed points and maximum values of this family of multimodal maps g2, g2.8571, and g4 are shown in Table 1.The cobweb diagrams of the maps: gγ3=4:I⟶I, see Figure 1(a); gγ2=2.8571:0,0.5⟶0,0.5, see Figure 1(b); and gγ1=2:0,0.25⟶0,0.25 (see Figure 1(c)), using the (31) with x0=0.05. These maps gγi are surjective on Ji=0,ζi, with i=1,2,3; besides, these maps fulfill theorem (3). Then maps gγ1, gγ2 and gγ3 are transitive on 0,ζ1, 0,ζ2 and I, respectively, as is shown in Figure 1. Figure 2 shows the bifurcation diagram where it is possible to observe that period 2 is a route to chaos because the family is based on the logistic map.
The Lyapunov exponents of these sequences are 0.6932, 1.0127 and 1.3374 forγ1=2, γ2=2.85, and γ3=4, respectively. Then, the family composed of a unimodal map, a bimodal map, and a trimodal map is given by (31) exhibits chaotic behavior.Figure 1
Cobweb diagram of the maps: (a)gγ3=4:I⟶I, (b)gγ2=2.8571:0,0.5⟶0,0.5 and (c)gγ1=2:0,0.25⟶0,0.25, using equation (31) with x0=0.05 which a long evolution covers completely the intervals I, 0,ζ2 and 0,ζ1, for γ3, γ2, γ1, respectively.
(a)(b)(c)Table 1
Maximum values and fixed points of the multimodal familygγkk=13, for γ1=2, γ2=2.85, and γ3=4.
kgγkxc1gγkxc2gγkxc3x¯L,R1x¯L,R2x¯L,R310.250.350.50, 0.1875−,−−,−20.35630.50.71430, 0.20620.2949, 0.4238−,−30.50.710, 0.21870.2779, 0.44970.5899, 0.8475Figure 2
Bifurcation diagram of system (31).Example 4.
Analysis of a non-transitivity example, we define a multimodal family with the following parameters:m=3, ζii=03=0,1/4,1/2,1. ρii=13=0.26,0.51,1, which form a trimodal map given as follows:(32)xn+1=gγkxn=γk4.160.25−xnxn,for0≤xn≤0.25;8.160.50−xnxn−0.25,for0.25<xn≤0.50;41−xnxn−0.50,for0.50<xn≤1..
Notice that the values ofρii=13 do not fulfill Theorem 3 (c), only the conditions of Theorem 1 are fulfilled, maxi≤3ρi=1, ζ1/ρ1=0.9615<ζ2/maxj≤2ρi=0.9804<1. Then there exist control parameter values that form a multimodal family which are γ1=3.8462, γ2=3.9216 and γ3=4, see Figure 3(a)–3(c)), respectively. Meanwhile, fixed points and maximum values of this family of multimodal familygγii=13 are shown in Table 2.
Each mapgγi is surjective on Ji=0,ζi, with i=1,2,3; however, all of them are not transitive because Theorem 3 is not fulfilled. Then the maps gγ2, gγ3 are not transitive on 0,ζ2 and I, respectively. Only the map gγ1 is transitive, see Figure 3(c).
Figure4 shows the bifurcation diagram of system (32), where it is possible to observe that this bifurcation diagram resembles the bifurcation diagram of the logistic map. It is worth mentioning that the initial condition used to compute the bifurcation diagram is x0=0.3. Multistability was numerically observed for this system, however, multistability is not addressed in this work because it is off target.Figure 3
Cobweb diagrams of the mapsgγii=13 given by equation (32) with x0=0.9 for: (a)γ3=4, (b)γ2=3.9216 and (c)γ1=3.8462.
(a)(b)(c)Table 2
Maximum values and fixed points of the multimodal familygγkk=13, for γ1=3.846, γ2=3.922, and γ3=4.
kgγkxc1gγkxc2gγkxc3x¯L,R1x¯L,R2x¯L,R310.250.490.9620, 0.1880.296, 0.4220.596, 0.83920.2550.50.980, 0.1890.295, 0.4240.593, 0.84330.260.5110, 0.190.294, 0.4260.59, 0.848Figure 4
Bifurcation diagram of system (32) by using x0=0.3.


## 5. Numerical Example of Trimodal Family

Now, in this Section5 we use the developed theory of this class of discrete maps to create a trimodal family which is comprised by a unimodal map, a bimodal map and a trimodal map. Consequently, a procedure to construct multimodal family of maps is given and a numerical example is also provided as follows: Select m parameters to generate a partition of the interval I,ζi<i=0m (where ζ0=0<ζ1⋯<ζm=1). Set the parameter values ρii=1m taking into account that Theorems 1 and 3 are fulfilled, and select the control parameters γk outside the regions where there exists asymptotically stable fixed points.Example 5.
Now, we give an example using the aforementioned steps to design a multimodal family based on a unimodal map, bimodal map, and a trimodal map. Som=3, and we select ζi=03=0,1/5,1/2,1.
We want thatgγ3xc1>gγ3xc2<gγ3xc3, but gγ3xc1>gγ3xc3, then ρ1=1 and ρ1/σ12=25. Therefore, we define gγ3xn in the subinterval 0,1/5 as follows:(33)xn+1=γk⋅251/5−xnxn−0.
The fixed pointsx¯L1 and x¯R1 are given by 0 and 0.1900, respectively. Because ρ1=1 and ζ1≠ζ2≠ζ3, then Theorem1 is fulfilled. We have gγ3xc1>gγ3xc2 then ρ1=1>ρ2, so we set ρ2=0.91 that generates ρ2/σ22=10.1111. Therefore, we have that gγ3xn is defined in the subinterval (1/5,1/2] as follows:(34)xn+1=γk⋅10.11111/2−xnxn−1/5.
The fixed pointsx¯L2 and x¯R2 are given by 0.2193, and 0.4560, respectively. Due to g42xc2∉S2 then Theorem 3 b) is fulfilled. Now we want that ρ2<ρ3<1, so we set ρ3=0.95 and ρ3/σ32=3.8. Therefore, gγ3xn is defined in the subinterval 1/2,1 as follows:(35)xn+1=γk⋅3.81−xnxn−1/2,the fixed points x¯L3 and x¯R3 are given by 0.5978 and 0.8364, respectively. Because g42xc3=0.3420<xL3=0.5978 and x¯L3<g4xc2 then Theorem 3 b) and c) is fulfilled. So ρii=13=1,0.91,0.95 is given and determine different modal values for a not uniform partition on I. Thus ζi=03 and ρii=13 define a trimodal map as follows:(36)xn+1=gγkxn=γk250.2−xnxn−0,for0≤xn≤0.2;10.11110.5−xnxn−0.2,for0.2<xn≤0.5;3.81−xnxn−0.5,for0.5<xn≤1..
The control parameter values for a multimodal family areγ1=4ζ1=0.8, γ2=4ζ2=2, γ3=4. In Table 3, the maximum values and fixed points of the multimodal family gγkk=13 are given. The control parameter values where the fixed points are asymptotically stable are given as follows(37)γk∈Υ=0,0.2∪0.2,0.6∪1.4642,1.7272∪3.0676,3.4022.
Since none of the control parameter values of the multimodal family
gγkk=13 belongs to the above interval Υ, then there are not asymptotic stable fixed points. Now, we only need to verify transitivity for g2, because for the unimodal case g0.8 the map is transitive due to it is unimodal. For γ=2, we have g22xc2=0.2320<x¯L2=0.2500, therefore Theorem 3 is fulfilled. Figure 5 shows a cobweb diagrams under the map: (a) xn+1=gγ1=0.8xn, (b) xn+1=gγ2=2xn, and (c) xn+1=gγ3=4xn.
Figure6 shows three time series obtained from trimodal map calculated with (36): (a) g0.8, (b) g2, and (c) g4, with x0=0.127.
The maps of the multimodal family are iterated up to one million data are generated, where each datum (with 16 significant digits) is distributed in the interval0,1. Afterwards, the Lyapunov exponents of these sequences are computed. For these numerical computation, more than 2000 initial values are employed for each map and, we do not find an orbit that has no positive Lyapunov exponent. Then the family comprise by a unimodal map, a bimodal map, and a trimodal map is given by (36) and exhibit chaotic behavior.
Figure7 shows the bifurcation diagram of the system given by (36), where it is possible to observe that this bifurcation diagram resembles the bifurcation diagram of the logistic map. It is worth mentioning that the initial condition used to compute the bifurcation diagram is x0=0.127.
In the literature, others piecewise multimodal maps have been reported that show regular partition intervals and use the same height for all modals. For example, in reference [25] a multimodal map and its basin of attraction were presented, where all modals have the same height and only numerical results were presented. Also in [27] the authors derived analytic expressions for the autocorrelation function and the auto-spectral density function of chaotic signals generated by a multimodal skew tent map and all modals with the same height.Table 3
The maximum values and fixed points of the multimodal familygγkk=13, with γ1=0.8, γ2=2, and γ3=4.
kgγkxc1gγkxc2gγkxc3x¯L,R1x¯L,R2x¯L,R310.20.180.190, 0.15−−20.50.450.4750, 0.180.2491, 0.4015−310.910.950, 0.190.2193, 0.45600.5978, 0.8364Figure 5
Cobweb Diagrams obtained from trimodal family equation (36) for initial condition x0=0.127: (a)gγ1=0.8xn; (b)gγ2=2xn; and (c)gγ3=4xn.
(a)(b)(c)Figure 6
Time series obtained from trimodal map calculated with equation (36): (a)g0.8, (b)g2, and (c)g4, with x0=0.127.
(a)(b)(c)Figure 7
Bifurcation diagram of system (36) by using x0=0.127.


## 6. Conclusions

In this work, the concept of multimodal family of m-modal maps based on the logistic map was introduced, we also gave the necessary and sufficient conditions to build a multimodal family in regular and irregular intervals. These families can display a unimodal map, bimodal map, up to m-modal map. The generation of a multimodal family of m-modal maps is warranty by means of the transitivity property. Also the stability of the fixed points was analyzed.This work could be continued in applications about pseudo-random bit generators to use them in the development of cryptographic systems.



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