Dynamics of Multimodal Families of m-Modal Maps

In this work, we introduce families of multimodal maps based on logistic map, i.e., families of m-modal maps are deﬁned on an interval I ⊂ 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.


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][5][6]; some applications of these generators are in video [7,8], secure communications [9,10], and cryptographic systems [11][12][13][14][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: 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. e 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. e useful insights for the study of chaos can be known by analyzing one-dimensional (1D) maps, e.g., the logistic and tent maps. ese systems have been extensively studied [28] and implemented experimentally [29][30][31][32][33][34]. A unimodal map is a continuous 1D function R ⟶ R with a single critical point c 0 , monotonically increasing on one side of c 0 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].
where N(t) is the state of the system at time t, α g is the intrinsic growth rate, and c c 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. e 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. e development of this new class of families of chaotic map are based on piecewise continuous function. e 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. e article is organized as follows: In Section 2, 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.

m-Modal Maps
Let (I, d) be the compact metric space I � [0, 1] endowed with the Euclidean metric. It is given the next definition: Definition 1. Let S and f be an interval and a function, respectively, such that S ⊂ I and f ∈ C(S, I). If there exists an x c ∈ S such that for all x ∈ S we have that f| x<x c is strictly increasing and f| x c < x is strictly decreasing then f is called a unimodal map on S.
Logistic map and tent map are examples of unimodal maps (I � S). A bimodal map can be defined by considering I � S 1 ∪ S 2 , S 1 ∩ S 2 � ∅, such that the map f is a unimodal map in each interval S 1 ⊂ I and S 2 ⊂ I. erefore, 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. e 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: where x is the state variable; c ∈ R is a bifurcation parameter, and α, β are fixed arbitrary parameters with restricted values 0 ≤ α < β ≤ 1. e first and second derivatives of f c are f c ′ � c(−2x + α + β) and f c ″ � −2c, respectively, then f c is continuous for all x ∈ R and has a local maximum or a local minimum at x c � (α + β)/2, (f c ′ (x c ) � 0), for c ≠ 0. erefore, to get a unimodal map on the interval (α, β) based on the logistic map, it is necessary to consider c > 0(c ∈ R + ).
In the case of m-modal maps based on a unimodal map, the interval I 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: { },such that ζ 0 � 0 < ζ 1 < · · · < ζ m � 1:

Complexity
If there exists at least a pair of subintervals S i and S j , with { }, 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 on I, where Now, a m-modal map is defined based on m unimodal maps as follows. { },respectively. If the map g is unimodal on each S i ∈ 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 S i 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 erefore, the continuous piecewise function is defined as follows: Equation (4) defines a map with m modals, where c is the bifurcation parameter. Recall that the modal of each subinterval S i is given by x (i) c � (ζ i−1 + ζ i )/2, for i � 1, . . . , m. en, the local maximum of the map (4) is { } and the local maximum of each unimodal map in Equation (4) is the same given by c/(4m 2 ).But if Π is nonuniform, then there exists at least a pair of subintervals S i with different diameter σ i ≠ σ j ; therefore, for a given value of c, the m-modal map (4) presents multiple local maximums h( Notice that the local maximums are determined by the parameter c and the diameter σ i of each subinterval S i . e interest is to control the local maximums independently of each σ i , so a new parameter ρ i is considered in Equation (4). To avoid the effect of the parameter σ i to determine the local maximum, then we consider ρ i /σ 2 i in each piece of the m-modal map (4). A m-modal map is given as follows: where ρ i > 0, with i � 1, . . . , m, are arbitrary fixed parameters but restricted to g c :

Lemma 1. Maximum ρ -value:
Consider a m-modal map given by (5). If g 4 : I ⟶ I, then at least one ρ i � 1 and the Proof. For c � 4, the local maximums are given by ρ 1 , ρ 2 , . . . , ρ m . Because g 4 : I ⟶ I then at least one ρ i � 1 and none of them is greater than 1. If the ρ's are different ρ 1 ≠ ρ 2 ≠ · · · ≠ ρ m , then only one ρ i � 1 and the others ρ i ∈ (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 is configuration allows to have at least one ρ i � 1, and the others ρ i ∈ (0, 1]. e proof is completed.

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 parameter c. In other words, these monoparametric families g c behave like a k-modal map, with k � 1, . . . , m, if it is possible to choose the appropriate values c 1 , c 2 , . . . , c m for the bifurcation parameter c which makes g c k behaves like a unimodal map, a bimodal map, up to a m-modal map, respectively. e property is that ∪ k i�1 S i ⊂ I will be invariant under each g c k such that arbitrary length orbits could be calculated from iterating any initial value x 0 ∈ I, i.e., for each c k , the interval [0, ζ k ] would be invariant under g c k .
Definition 4. Let g c k ∈ C(I, R) be a continuous m-modal map given by (5) should determine the existence of parameters c k . So, in the next theorem, necessary and sufficient conditions are given to construct multimodal families by using m-modal maps. Considering that for c � 4, g c is surjective on I.
Theorem 1 (configurations). Let g c (x) be a m-modal map given by equation (5) and for every j ∈ 1, 2, . . . , m − 1 { }: Proof. (⇐) We have that g 4 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 As g 4 is a m-modal map function given by (5), then each subinterval of the partition Π � S 1 , S 2 , . . . , S m on I satisfies that g 4 (ζ i ) � 0, g 4 (x (i) c ) � ρ i and ρ i > 0, due to they are local maximums. According to Corollary 1, as g 4 is surjective on I there exists at least a local maximum ρ i � 1 and it is also satisfied that every ρ i ≤ 1. e local maximum under g c on each subinterval of Π is given by Since for each local maximum, the parameter value of c is the same, i.e., it is a constant, then the maximum value of g c restricted to the interval We are finding a parameter value c k ∈ (0, 4) such that max x∈J k g c k (x) � ζ k . Doing some algebra, it results that Due to (7), c k < 4, for k ∈ 1, 2, . . . , m − 1 { }, and the local As g c k is continuous on J k then it is satisfied g c k is surjective on J k � [0, ζ k ]; besides, as k is arbitrary then there exist m control parameters (10) and (6) we have: m); therefore the family g c k is a multimodal family of a m-modal maps.(⇒) Now, we have that g c m k�1 , for c ∈ (0, 4], is a multimodal family of m-modal maps given by (5), so g 4 is surjective on I and there exists c k such that g c k : [0, ζ k ] ⟶ [0, ζ k ], for k � 1, . . . , m. We need to prove that the existence of c k implies that equations (6) and (7) are fulfilled.
We know that for each c k there are k local maximums at Due to c k ∈ (0, 4), for k � 1, . . . , m − 1 and c m � 4, then c k < c m , for k � 1, . . . , m − 1. So c k /4 < c m /4 � 1, therefore which fulfills equation (7) and the proof is completed. . With the aforementioned parameters, a trimodal map is defined by (5) as follows:

□
4 Complexity because max j≤3 ρ i � 1, then the trimodal map g 4 is surjective on I. Besides, it is easy to check that the trimodal map given by (15) fulfills the equation (6) and (7)  Proof.
is corollary is a direct consequence of eorem 1.

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.

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). Let g c (x) be a m-modal map given by equation (5) on I if the following interval exists
and c j ∈ Υ j , then two fixed points of system (5) exist in each subinterval [ζ j−1 , ζ j ] and at least one is stable.
Proof. We start by calculating the fixed points of a multimodal family with m maps g c j m j�1 , then the following equation is solved where g c k is a multimodal map given by (5) and defined in m . e fixed points are calculated by the following equation 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. e fixed points are given as follows: ere are two fixed points if the discriminant is positive. c k > 0 for each mutimodal family, thus the fixed points exist if the following inequality is fulfills: Now, we need to prove that at least one of the fixed points is stable. In our case x e proof is completed. e inequality (20) warranties the existence of two fixed points in each interval S i , with i � 1, . . . , m. However, when g c j domain is restricted to the interval S 1 � [ζ 0 � 0, ζ 1 ], there always exists the fixed point x (1) L � 0 which is asymptotically stable when the condition |g c 1 ′ (x (1) L )| < 1 is fulfilled.
is results in c 1 ∈ (0, ζ 1 /ρ 1 ), then x (1) L � 0 is asymptotically stable and is unique. When c k > ζ 1 /ρ 1 there exist two fixed points and the second fixed point is given by is fixed point x (1) R is asymptotically stable always that c 1 ∈ (ζ 1 /ρ 1 , 3ζ 1 /ρ 1 ) is fulfilled, according to (21). e set Υ is given by the union of the intervals [0, ζ 1 /ρ 1 ] and Υ j , if Υ j exists. us, Υ is called the stable set of c parameter values. erefore, if c ∈ Υ, then the m-modal map g c (x) given by (5) on I presents at least one stable fixed point.

Definition 6.
e set Υ * � [0, 4] − Υ is called the set of c parameter values such that the m-modal map g c (x) given by Equation (5) on I presents unstable fixed points, where Υ is the closure of Υ.
If any c k of the multimodal family g c 1 , g c 2 , . . . , g c m belongs to the set Υ, then the data series from the multimodal map are useless to generate chaotic behaviour.

Transitivity.
One of the important characteristics of dynamical systems is the transitivity property, which describes that given any open subsets U 1 , U 2 ∈ X, there exists an x 0 ∈ U 1 , and n > 0, such that f n (x 0 ) ∈ U 2 . It is worth to mention that eorem 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 g c k m k�1 on the interval J � [0, ζ k ], after the parameter values ζ i and ρ i are selected, so the transitivity is warrantied by the next proposition: and x (j) (26) and the following inequality is always preserved

Proof. See appendix A.
If the m-modal map presents combined modals, i.e., , with j � 1, . . . , m − 2, then eorem 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 maps g c i 3 i�1 , such that eorem 3 (c) is fulfilled. erefore, the parameter m � 3 determines that the interval I is partitioned in three subintervals. Arbitrarily, we propose the following parameters to generate the partition: is defined in the subinterval (0.5, 1] as follows: the fixed points x (3) L and x (3) R are given by 0.5899 and 0.8475, respectively. Accordingly eorem 3 c)0.5899<ρ 2 < 1, so we set ρ 2 � 0.7 that generates ρ 2 /σ 2 2 � 11.2. Now g c 3 (x n ) is defined in the subinterval (0.25, 0.5] as follows: the fixed points x (2) L and x (2) R are given by 0.2779 and 0.4497, respectively. en 0.2779<ρ 1 < 0.7, so ρ 1 � 0.5 and ρ 1 /σ 2 2 � 8. erefore, g c 3 (x n ) is defined in the subinterval [0, 0.25] as follows: the fixed points x (2) L and x (2) R are given by 0 and 0.2187, respectively. erefore, ρ i 3 i�1 � 0.5, 0.7, 1 { } forms a trimodal map given as follows: 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. e Lyapunov exponents of these sequences are 0.6932, 1.0127 and 1.3374 for c 1 � 2, c 2 � 2.85, and c 3 � 4, respectively. en, the family composed of a unimodal map, a bimodal map, and a trimodal map is given by (31) exhibits chaotic behavior.  Table 2.

Complexity
Each map g c i is surjective on J i � [0, ζ i ], with i � 1, 2, 3; however, all of them are not transitive because eorem 3 is not fulfilled. en the maps g c 2 , g c 3 are not transitive on [0, ζ 2 ] and I, respectively. Only the map g c 1 is transitive, see Figure 3(c). Figure 4 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 x 0 � 0.3. Multistability was numerically observed for this system, however, multistability is not addressed in this work because it is off target.

Numerical Example of Trimodal Family
Now, in this Section 5 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 (where ζ 0 � 0 < ζ 1 · · · < ζ m � 1). Set the parameter values ρ i m i�1 taking into account that eorems 1 and 3 are fulfilled, and select the control parameters c k outside the regions where there exists asymptotically stable fixed points.

(37)
Since none of the control parameter values of the multimodal family g c k 3 k�1 belongs to the above interval Υ, then there are not asymptotic stable fixed points. Now, we only need to verify transitivity for g 2 , because for the unimodal case g 0.8 the map is transitive due to it is unimodal. For c � 2, we have g 2 2 (x (2) c ) � 0.2320 < x (2) L � 0.2500, therefore eorem 3 is fulfilled. Figure 5 shows a cobweb diagrams under the map: Figure 6 shows three time series obtained from trimodal map calculated with (36): (a) g 0.8 , (b) g 2 , and (c) g 4 , with x 0 � 0.127. e 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 interval (0, 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 Table 3: e maximum values and fixed points of the multimodal family g c k 3 k�1 , with c 1 � 0.8, c 2 � 2, and c 3 � 4.

Complexity 9
find an orbit that has no positive Lyapunov exponent. en the family comprise by a unimodal map, a bimodal map, and a trimodal map is given by (36) and exhibit chaotic behavior. Figure 7 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 x 0 � 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.

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. ese families can display a unimodal map, bimodal map, up to m-modal map. e 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.
is work could be continued in applications about pseudo-random bit generators to use them in the development of cryptographic systems.  10 Complexity