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 interval , which is partitioned into non-uniform subdomains, with . 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 -modal map. Some numerical examples are given according to the developed theory. Some numerical examples are given in accordance with the developed theory.
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 . 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 .
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 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 with a single critical point , monotonically increasing on one side of 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 is the state of the system at time , is the intrinsic growth rate, and 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 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 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
Let be the compact metric space endowed with the Euclidean metric. It is given the next definition:
Definition 1. Let and be an interval and a function, respectively, such that and . If there exists an such that for all we have that is strictly increasing and is strictly decreasing then is called a unimodal map on .
Logistic map and tent map are examples of unimodal maps . A bimodal map can be defined by considering , , such that the map is a unimodal map in each interval and . Therefore, there is a map with two modals on , called a bimodal map on . 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:where is the state variable; is a bifurcation parameter, and , are fixed arbitrary parameters with restricted values . The first and second derivatives of are and , respectively, then is continuous for all and has a local maximum or a local minimum at , , for . Therefore, to get a unimodal map on the interval based on the logistic map, it is necessary to consider .
In the case of m-modal maps based on a unimodal map, the interval 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 as follows:
Definition 2. Let be a partition of which is determined by a finite sequence , with ,such that :If there exists at least a pair of subintervals and , with , such that , where is the diameter of , with , then the partition on is called nonuniform. In the contrary case, the partition on is called uniform.
To illustrate the above definition, it is given the following example.
Example 1. Consider the following sequence of points on ,
, we have which determine the partition of , , where , therefore, the partition is nonuniform.
Now, a m-modal map is defined based on unimodal maps as follows.
Definition 3. Let and be a map and a partition on with ,respectively. If the map is unimodal on each , with , 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 of the partition of . To determine the piecewise function that defines the m-modal map, we start by giving the sequence of points , which determine a partition on . Therefore, the continuous piecewise function is defined as follows:Equation (4) defines a map with modals, where is the bifurcation parameter. Recall that the modal of each subinterval is given by , for . Then, the local maximum of the map (4) is given by ,in each . In the case that the partition on is uniform, each with and the local maximum of each unimodal map in Equation (4) is the same given by .But if is nonuniform, then there exists at least a pair of subintervals with different diameter ; therefore, for a given value of , the m-modal map (4) presents multiple local maximums . Notice that the local maximums are determined by the parameter and the diameter of each subinterval . The interest is to control the local maximums independently of each , so a new parameter is considered in Equation (4). To avoid the effect of the parameter to determine the local maximum, then we consider in each piece of the m-modal map (4). A m-modal map is given as follows:where , with , are arbitrary fixed parameters but restricted to , for , for all .
Lemma 1. Maximum -value: Consider a m-modal map given by (5). If , then at least one and the others , with .
Proof. For , the local maximums are given by . Because then at least one and none of them is greater than 1. If the ’s are different , then only one and the others , with . If all ’s are not different then this allows the possibility of having more . However, they can be always ordered as , with . This configuration allows to have at least one , and the others . 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 parameter . In other words, these monoparametric families behave like a -modal map, with , if it is possible to choose the appropriate values for the bifurcation parameter which makes behaves like a unimodal map, a bimodal map, up to a m-modal map, respectively. The property is that will be invariant under each such that arbitrary length orbits could be calculated from iterating any initial value , i.e., for each , the interval would be invariant under .
Definition 4. Let be a continuous m-modal map given by (5), where ,if there exist different values for the bifurcation parameter such that they fulfill(i) is surjective, with .The monoparametric family is called multimodal family of m-modal maps.
The above definition allows us to see that to obtain a multimodal family, parameter values of , and should determine the existence of parameters . So, in the next theorem, necessary and sufficient conditions are given to construct multimodal families by using m-modal maps. Considering that for , is surjective on .
Theorem 1. (configurations). Let be a m-modal map given by equation (5) on with . There exist control parameter values such that the family is a multimodal family of m-modal map if and only if:and for every :
Proof. We have that is a m-modal map and is sujerctive on and suppose that conditions given by equations (6) and (7) are fulfilled; it must be proved that there exist different control parameter values , such that are surjective, i.e., , where .
As is a m-modal map function given by (5), then each subinterval of the partition on satisfies that , and , due to they are local maximums. According to Corollary 1, as is surjective on there exists at least a local maximum and it is also satisfied that every .
The local maximum under on each subinterval of is given byThus, when is restricted to the interval , then there are local maximums , , , , with . Since for each local maximum, the parameter value of is the same, i.e., it is a constant, then the maximum value of restricted to the interval isWe are finding a parameter value such that . Doing some algebra, it results thatDue to (7), , for , and the local maximum of on isAs is continuous on then it is satisfied is surjective on ; besides, as is arbitrary then there exist control parameters such that are surjective, with . By equations (10) and (6) we have:so for all . Then there are different values for the bifurcation parameter : and is invariant under each where ; therefore the family is a multimodal family of a m-modal maps.
Now, we have that , for , is a multimodal family of m-modal maps given by (5), so is surjective on and there exists such that , for . We need to prove that the existence of implies that equations (6) and (7) are fulfilled.
We know that for each there are local maximums at , , , . Because is surjective on , then it is satisfied that . As a consequence and . So the control parameter value is given by , then . We also know that there are different values of : . Since dividing the values by four does not affect the inequalities, then the condition (6) is fulfilled.Due to , for and , then , for . So , thereforewhich fulfills equation (7) and the proof is completed.
Example 2. As an example, it is selected and which determine a nonuniform partition of and the local maximum for each subinterval . With the aforementioned parameters, a trimodal map is defined by (5) as follows:because , then the trimodal map is surjective on . 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 , , and . Therefore, given by (15) is a multimodal family comprises by a unimodal map, bimodal map and trimodal map, i.e., defines a unimodal map, defines a bimodal map, and defines a trimodal map. It is worth mentioning that the trimodal map does not have a chaotic behavior for all the considered values of ’s. The Lyapunov exponents are 0.6931, -1.0312, and 1.1162 for , , and , respectively. Note that for 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 ’s. Therefore, we need to consider values of such that the multimodal families present unstable fixed points.
Corollary 1. (uniform partition). Let be a m-modal map given by equation (5) on with , uniform partition , and . Then there exist control parameter values such that the family is a multimodal family of m-modal map.
Proof. This corollary is a direct consequence of Theorem 1.
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). Let be a m-modal map given by equation (5) on if the following interval existsand , then two fixed points of system (5) exist in each subinterval and at least one is stable.
Proof. We start by calculating the fixed points of a multimodal family with maps , then the following equation is solvedwhere is a multimodal map given by (5) and defined in intervals , , , . The fixed points are calculated by the following equationwhere . Each part of the function must have a maximum of two fixed points; thus an m-modal map must have a maximum of fixed points. The fixed points are given as follows:There are two fixed points if the discriminant is positive. 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 is stable if for defined in which is satisfied whenThe proof is completed.
The inequality (20) warranties the existence of two fixed points in each interval , with . However, when domain is restricted to the interval , there always exists the fixed point which is asymptotically stable when the condition is fulfilled.This results in , then is asymptotically stable and is unique. When there exist two fixed points and the second fixed point is given byThis fixed point is asymptotically stable always that is fulfilled, according to (21).
Definition 5. The set is given by the union of the intervals and , if exists. Thus, is called the stable set of parameter values.
Therefore, if , then the m-modal map given by (5) on presents at least one stable fixed point.
Definition 6. The set is called the set of parameter values such that the m-modal map given by Equation (5) on presents unstable fixed points, where is the closure of .
If any of the multimodal family belongs to the set , 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 subsets , there exists an , and , such that . 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 on the interval , after the parameter values and are selected, so the transitivity is warrantied by the next proposition:
Theorem 3. Let be a multimodal family of m-modal maps given by equation (5) on , with . Then the multimodal family of m-modal maps is transitive for . If any of the following cases occur:
(a)(b)and for all , if there exist and in the interval.
(c)and the following inequality is always preservedwhere and are the critical point and the left fixed point in the -th interval, with , and is the fixed point in the interval.
Proof. See appendix A.
If the m-modal map presents combined modals, i.e.,
, with , 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 maps , such that Theorem 3 (c) is fulfilled. Therefore, the parameter determines that the interval is partitioned in three subintervals. Arbitrarily, we propose the following parameters to generate the partition: . We also know that , then and . Therefore, is defined in the subinterval as follows:the fixed points and are given by 0.5899 and 0.8475, respectively. Accordingly Theorem 3 c)0.5899<ρ2 < 1, so we set that generates . Now is defined in the subinterval as follows:the fixed points and are given by 0.2779 and 0.4497, respectively. Then 0.2779<ρ1 < 0.7, so and . Therefore, is defined in the subinterval as follows:the fixed points and are given by 0 and 0.2187, respectively. Therefore, forms a trimodal map given as follows:Due to , , conditions of Theorem 1 are fulfilled, then there exist control parameter values that form a multimodal family which are , and , see Figure 1. Meanwhile fixed points and maximum values of this family of multimodal maps , , and are shown in Table 1.The cobweb diagrams of the maps: , see Figure 1(a); , see Figure 1(b); and (see Figure 1(c)), using the (31) with . These maps are surjective on , with ; besides, these maps fulfill theorem (3). Then maps , and are transitive on , and , 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 for , , and , respectively. Then, the family composed of a unimodal map, a bimodal map, and a trimodal map is given by (31) exhibits chaotic behavior.
(a)
(b)
(c)
Example 4. Analysis of a non-transitivity example, we define a multimodal family with the following parameters: , . , which form a trimodal map given as follows:Notice that the values of do not fulfill Theorem 3 (c), only the conditions of Theorem 1 are fulfilled, , . Then there exist control parameter values that form a multimodal family which are , and , see Figure 3(a)–3(c)), respectively. Meanwhile, fixed points and maximum values of this family of multimodal family are shown in Table 2.
Each map is surjective on , with ; however, all of them are not transitive because Theorem 3 is not fulfilled. Then the maps , are not transitive on and , respectively. Only the map 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 . Multistability was numerically observed for this system, however, multistability is not addressed in this work because it is off target.
(a)
(b)
(c)
5. 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 parameters to generate a partition of the interval (where ). Set the parameter values taking into account that Theorems 1 and 3 are fulfilled, and select the control parameters 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. So , and we select .
We want that , but , then and . Therefore, we define in the subinterval as follows:The fixed points and are given by 0 and 0.1900, respectively. Because and , then Theorem1 is fulfilled. We have then , so we set that generates . Therefore, we have that is defined in the subinterval as follows:The fixed points and are given by 0.2193, and 0.4560, respectively. Due to then Theorem 3 b) is fulfilled. Now we want that , so we set and . Therefore, is defined in the subinterval as follows:the fixed points and are given by 0.5978 and 0.8364, respectively. Because and then Theorem 3 b) and c) is fulfilled. So is given and determine different modal values for a not uniform partition on . Thus and define a trimodal map as follows:The control parameter values for a multimodal family are , , . In Table 3, the maximum values and fixed points of the multimodal family are given. The control parameter values where the fixed points are asymptotically stable are given as followsSince none of the control parameter values of the multimodal family
belongs to the above interval , then there are not asymptotic stable fixed points. Now, we only need to verify transitivity for , because for the unimodal case the map is transitive due to it is unimodal. For , we have , therefore Theorem 3 is fulfilled. Figure 5 shows a cobweb diagrams under the map: (a) , (b) , and (c) .
Figure 6 shows three time series obtained from trimodal map calculated with (36): (a) , (b) , and (c) , with .
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 interval . 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.
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 .
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.
(a)
(b)
(c)
(a)
(b)
(c)
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.
Appendix
A. Transitivity
Proof (a). ⟹We need to prove that any open subsets , , there exists an , and , such that .
For the case , we have , with and the critical point divides this interval in two intervals and . Therefore, we have and . Then there are two points and , , that divide the intervals and , respectively, such that , , , and , with , and .
The intervals and contain points , and , respectively, such that , and each interval , with i = 1,…,4, is divided into two, generating eight subintervals . Now we have that each of these intervals are mapped by , with .
Notice that the points ’s always exit in the intervals ’s. We can continue up to any of the intervals is contained in . Suppose that the interval is contained in , then this interval is mapped to the whole interval , , with . This implies that there exists an , and , such that . The dynamical system is transitive.
For the case , there are two critical points in , i.e., and , and . The critical points and divide these intervals into intervals , , , and , such that , with , then there are four points , with that divide the intervals , respectively, such that , with , with , . , .
Now, the intervals , with , contain points , with , respectively, such that each interval is divided into two, generating sixteen subintervals , with . Now we have that each of these intervals maps , with .
Notice that the points ’s always exist in the intervals ’s. We can continue up to any of the intervals is contained in . Suppose that the interval is contained in , then this interval is mapped to the whole interval , , with . This implies that there exists an , and , such that . The dynamical system is transitive.
In general, for , we have and there are critical points , with that divide the intervals , with , into intervals such that , with . Therefore, there exist ’s that divide the intervals , with , into intervals such that , with . The points ’s always exist into the intervals ’s. We can continue up to any of the intervals , with , is contained in . Suppose that the interval is contained in , then this interval is mapped to the whole interval , , with . This implies that there exists an , and , such that . The dynamical system is transitive for the case (a).
(b)
We need to prove that any open subsets , with , there exists an , and , such that . For the case , we have , with . This case is proved in the same way that the previous case (a) for .
For the case , , with , and there are two critical points and , with and . The critical points and divide the intervals and , respectively, into four intervals , , , . Therefore, we have , with , and , with . Firstly, we analyze the cases , and later cases .
For , there are and such that . Then it is possible to generate four intervals , , , and . Each of these intervals fulfills , with . This implies that there are such that , with . Then it is possible to generate eight intervals such that , with . Once again, the ’s always exist because we have that each of the intervals fulfills , with . The refinement of the intervals continues up to any of the intervals is contained in . Suppose that the interval is contained in , then this interval is mapped to the whole interval , , with . This implies that there exists an , and , such that .
For the cases , . There are three cases: ; ; and .
For the first case, if , then . We know that there exist intervals , with , such that each interval fulfills . If , then the diameter of each interval and each of these intervals has four preimages . Two in and two in . So, if then . This implies that there exists an , and , such that . Therefore, is transitive in for . The second case . There are points such that , and , then the intervals and are mapped to the whole interval . Therefore, these two intervals can be considered as the first case, however, the other interval . , such that , with because for all . There exists a such that and , then at least once any point has been mapped to the interval . This implies that there exists an , and , such that . Therefore, is transitive in for . For the third case , there are two unstable fixed points , with . There are two intervals and , with that fall in the previous case because , for all . Therefore, is transitive in for . The middle interval contains an interval , with , such that , so this case falls in the previous case. For the intervals and , we have , for some . This implies that there exists an , and , such that . Therefore, is transitive in for .
The general case , because , with and for all , then there are such that , with . And is always transitive in because . This implies that there exists an , and , such that . The dynamical system is transitive for the case (b).
(c) For the case , we have , with . This case is proved in the same way that the previous case (a) for .
For the case , , with , and there are two critical points and , with and . There are two points such that , then . For all there is a such that , so , and the same for all . Therefore, for all there is a such that . Now, to prove transitivity, we only need to show that there exist intervals of whatever tiny diameter in that are mapped by to the whole interval .
The critical point divides the interval , into two intervals , and . So, we have , with , because . There are and such that . Then it is possible to generate four intervals , , , and . Each of these intervals fulfills , with . This implies that there are such that , with . Then it is possible to generate eight intervals such that , with . Once again, the ’s always exist because we have that each of the intervals fulfills , with . The refinement of the intervals continues up to any of the intervals is contained in . Suppose that the interval is contained in , then this interval is mapped to the whole interval , , with . For all is mapped to then there is a such that , so there are preimages of the such that , with . This implies that there exists an , and , such that .
We prove an arbitrary case . Then the general case is a direct consequence of this case because , with , and the following inequality is always preserved
Therefore, for all there is a such that . Then is transitive, for . This completes the proof.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
There are no conflicts of interest regarding the publication of this paper.
Acknowledgments
B. B. Cassal-Quiroga thanks CONACYT for the Ph.D. scholarship granted (register number 262247). E. Campos-Cantón acknowledges CONACYT for the financial support through project no. A1-S-30433.