The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

Li, Risong; Lu, Tianxiu

doi:https://doi.org/10.1155/2020/9379628

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9379628 | https://doi.org/10.1155/2020/9379628

The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

Risong Li^1,2and Tianxiu Lu^3,4

Academic Editor: Hassan Zargarzadeh

Received09 Jun 2020

Revised08 Aug 2020

Accepted09 Sept 2020

Published28 Sept 2020

Abstract

In this paper, we study some chaotic properties of -dimensional dynamical system of the form where for any is an integer, and is a compact subinterval of the real line for any . Particularly, a necessary and sufficient condition for a cyclic permutation map to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy of such a cyclic permutation map is the same as the topological entropy of each of the following maps: if and , and that is sensitive if and only if at least one of the coordinates maps of is sensitive.

1. Introduction

Let and be continuous maps on the compact subintervals and of the real line , and let be a continuous map defined by for any . These maps have been proposed to give a mathematical description of competition in a duopolistic market, called Cournot duopoly (see [1]). This is the reason why the above map is called a Cournot map, and the above two maps and are called reaction functions (that is, the maps and give laws to organize the production of some firms which are competitors in a market).

From [1, 2], we know that there are the so-called Markov perfect equilibria (MPE henceforth) processes, where the two players move alternatively such that each of them chooses the best reply to the previous action of another player. This occurs if the phase point belongs alternatively to the graphs of the reaction curves and . This condition can be satisfied if the initial condition is a point of a reaction curve. That is, (player 1 moves first) or (player 2 moves first). This follows from the fact that the union of the graphs of the two reaction function is trapping for .

Probably, the first paper which gives the concept of chaos in a mathematically rigorous way is that of Li and Yorke [3]. Since then many different rigorous notions of chaos have been proposed. Each of these concepts tries to describe some kind of unpredictability in the evolution of the system. The notion of Li–Yorke sensitivity (LY-sensitivity) was presented for the first time by Akin and Kolyada in [4]. Moreover, they introduced the notion of spatiotemporal chaos. A very important generalization is distributional chaos, proposed by Schweizer and Smítal [5], mainly because it is equivalent to positive topological entropy and some other concepts of chaos when restricted to some spaces (see [5, 6]). It is noted that this equivalence does not transfer to higher dimensions, e.g., positive topological entropy does not imply distributional chaos in the case of triangular maps of the unit square [7] (the same happens when the dimension is zero [8]). In [9], Wang et al. introduced the concept of distributional chaos in a sequence and showed that it is equivalent to Li–Yorke chaos (LY-chaos) for continuous maps of the interval. During the last years, many researchers paid attention to the chaotic behavior of Cournot maps (see [1, 2, 10–17]).

From [18], we know that if is an infinite metric space, then if a continuous map is transitive and has dense periodic points then it has sensitive dependence on initial conditions, which means that the third assumption in definition of chaos in sense of Devaney (D-chaos) is not necessary. By definition, it is clear that if a continuous map is D-chaotic, then it is chaotic in sense of Ruelle–Takens (RT-chaotic). It was proved that the converse is false (see [13]). It is well known that D-chaotic maps are LY-chaotic (see [19]) and that maps with positive topological entropy (h-chaotic) are also LY-chaotic maps (see [20]). However, there exist LY-chaotic interval maps with zero topological entropy (see [21]). And from Theorem 1.2 in [13], we know that there exist LY-chaotic maps which are not D-chaotic. For interval continuous maps, J. S. Canovas and M. Ruiz Marin had the particular cases established in Theorem 1.2 which is from [13].

Let and The set represents the union of the graphs of the two reaction function and is trapping for , i.e., . Canovas and Ruiz Marín called the set a MPE set for (see [13]). Moreover, they discussed and studied several chaotic properties (e.g., Devaney chaos, RT chaos, topological chaos, and Li–Yorke chaos) of Cournot maps and showed that it is not true that any chaotic property they considered satisfies the condition that is chaotic if and only if is chaotic. Recently, Lu and Zhu further investigated the dynamical properties of Cournot maps, and more precisely, for the maps , , and (see [15]). In [22], Linero Bas and Soler Lopez defined a cyclically permuted direct product map and studied the topics of (totally) topological transitivity and (weakly) topological mixing for cyclically permuted direct product maps by exploring the relationship between the dynamics of and that of the compositions , where , (is called to be cyclically permuted direct product maps), which is defined from the Cartesian product into itself, where are general topological spaces, each map is continuous, , and is a cyclic permutation of . In [23], Linero Bas and Soler Lopez established some results on transitivity for cyclically permuted direct product maps of the Cartesian product , where . Particularly, it was shown that, for , the transitivity of this map is equivalent to the total transitivity, and if , they obtained a splitting result for transitive maps. Also, they extended well-known properties of transitivity from interval maps to cyclically permuted direct product maps. To do it, they used the strong link between the map and the compositions . For cyclically permuted direct product maps, if and , these maps appears associated with certain economical model so called Cournot duopoly (see [2, 11–13], etc.). Even one can find them in age-structured population models, as in [24], where it is analyzed the Leslie model. For study on population models, we refer the reader to [24] and the references therein.

For chaotic maps, there have been many applications. Since Li and Yorke [3] introduced the term of chaos in 1975, chaotic dynamical systems were highly discussed and investigated in the literature (see [18, 25, 26] and the references therein) as they are very good examples of problems coming from the theory of topological dynamics and model and many phenomena from biology, physics, chemistry, engineering, and social sciences. Recently, some new 1D and 2D chaotic systems with complex chaos performance have been developed (see [27–35] and the references therein).

Motivated by [13, 15, 22, 23], we will deal with the dynamical properties of cyclic permutation maps:defined from a Cartesian product into itself, where each is a compact subinterval of the real line, and

. Clearly, cyclic permutation map is a cyclically permuted direct product map. Since is a direct product of interval maps, , with for any and , it makes sense to analyze the dynamics of in terms of these compositions . In particular, a necessary and sufficient condition for a cyclic permutation map to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is given. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is discussed. More precisely, for a cyclic permutation map,where

We obtain the following results:(1) is LY-chaotic if and only if is LY-chaotic for any (2) is h-chaotic if and only if is h-chaotic for any (3)If is RT-chaotic, then is RT-chaotic for any (4)If is D-chaotic, then is D-chaotic for any (5) is topologically transitive if and only if is D-chaotic if and only if is RT-chaotic(6)If is topologically transitive, then it is h-chaotic(7)If is D-chaotic, then is D-chaotic(8)If is RT-chaotic, then so is (9) is h-chaotic if and only if so is (10) is LY-chaotic if and only if so is

Our results extend some existing ones on two-dimensional dynamical systems. Also, it is shown that the topological entropy of such a cyclic permutation map is the same as the topological entropy of each of the following maps: , and that it is sensitive if and only if at least one of the following maps is sensitive: .

The interest of studying continuous cyclic permutation maps is as follows. Firstly, they are -dimensional maps whose dynamical behavior is close to that of one-dimensional maps (see [22, 23, 36, 37] and the references therein), where is any given integer, and their study could give us information about the dynamics of general -dimensional maps. Secondly, these maps are closely related with the model of an economic process called Cournot duopoly (see [22–24, 38]), the age-structured population models (see [24] and the references therein), and the cyclically permuted direct product maps (see [22, 23] and the references therein). Thirdly, the chaotic dynamics is widely used in nonlinear control, synchronization communication, and many other applications (see [27–35] and the references therein).

2. Preliminaries

Let be a metric space with metric .

A dynamical system or a continuous map is said to be(1)Transitive if for every pair of nonempty open sets and of , there exists a positive integer such that .(2)Mixing if for every pair of nonempty open sets and of , there exists a positive integer such that for every integer .(3)Sensitive if there is an such that whenever is a nonempty open set of , there exist points such that for some positive integer , where is called a sensitive constant of .(4)Chaotic in the sense of Ruelle–Takens (or RT-chaotic, for short) (see [39]) if it is both transitive and sensitive.(5)Chaotic in the sense of Li–Yorke (or LY-chaotic, for short) if there is an uncountable set such that for any with : where is the set of periodic points of and a point is called a periodic point of if there is an integer such that .(6)Chaotic in the sense of Devaney (or D-chaotic, for short) if is transitive and sensitive with , denotes the closure of the set .

A subset is said to be an -separated if for any with there is with . Let be the maximal possible cardinality of an -separated set. The topological entropy of (see [25, 40]) is defined as

Recall that for any integer . A dynamical system or a map is h-chaotic if .

Let be a compact interval of the real line for any and endowed with the product metric which is defined byfor any

For any continuous map on a metric space and any , the set of limit points of the sequence is the -limit set of which is written by . Then, if has no isolated points, then the map is transitive if there exists with . Write . Then, is the set of transitive points of .

3. Main Results

3.1. Relation between Some Chaotic Properties of a Continuous Cyclic Permutation Map and the Corresponding Properties of Every Coordinate Map

It is known in the frame of general topological spaces (and for general cyclic permutations) that if is transitive (even mixing or weakly mixing or totally transitive) then the associate compositions so are. This was done in [22]. For completeness, we give the proof of Lemma 1.

We need the following two lemmas.

Lemma 1. For a continuous cyclic permutation map,whereIf it is topologically transitive, then every coordinate map of is topologically transitive.

Proof. As the proof of the result in Lemma 1 is similar for any , for simplicity we only prove Lemma 1 for . Suppose that is topologically transitive and letwithwhere is the -limit set of under . By (3) in [36], hypothesis, the definition,One can easily prove thatNow, we show that ifthenAs is topologically transitive, is surjective for any . So, for any there is with . Let be a sequence of positive integers withBy the continuity of ,That is, . This implies thatSimilarly, one can proveBy the similar argument, we obtain that ifthenand that ifthenBy the above argument and the transitivity of , it follows that the case ,, and cannot happen. Thus, by the above argument, we haveThis means that , , and are topologically transitive.
However, it is not true that the transitivity of implies the transitivity of . In fact, we can construct examples of maps which are not transitive but the compositions are transitive. Let us show an example.

Example 1. Let with transitive but not totally transitive, that is, is not transitive (here, is the unit interval, but we can translate the example to arbitrary sets ). Since is not transitive, we can decompose the unit interval into , being compact intervals such that , with and such that, and mixing, (for details, consult, for instance, [26]). In this case, and taking into account that , we find that . Consequently, is not transitive. Due to the fact that , according to [23] (see Corollary B), we have that is transitive if and only if is totally transitive; since is not transitive, we deduce that is not transitive although its associated composition is. Nevertheless, in the two-dimensional case, when , and , the transitivity of implies the transitivity of , as it was proved in Lemma 2.1 in [13] (Canovas-Marin).
We note that the following lemma holds in the general setting of topological spaces: to this end, consult Lemma 7 in [23]. So, the proof of Lemma 2 is omitted here.

Lemma 2. For a topologically transitive cyclic permutation map,where for any . Then, one has

Theorem 1. For a cyclic permutation map,where

The following hold:(1) is LY-chaotic if and only if is LY-chaotic for any (2) is h-chaotic if and only if is h-chaotic for any (3)If is RT-chaotic, then is RT-chaotic for any (4)If is D-chaotic, then is D-chaotic for any (5)If is RT-chaotic for any and satisfies that there is an transitive point of for any such that then is RT-chaotic.(6)If is D-chaotic for any and satisfies that there is an transitive point of for any such that Then, is D-chaotic.(7)If is RT-chaotic for any and satisfies that, for any and any , where means that and ; then, is not RT-chaotic.(8)If is D-chaotic for any and satisfies that, for any and any ,where means that and , then is not D-chaotic.

Proof. By Propositions 3.1 and 3.2 in [15], the definition, hypothesis, andOne can easily verify that statement (1) is true. By [41], the definition, and we can easily prove that (2) is true. Now, we show that statement (4) is true. Suppose that is D-chaotic, that is, it is topologically transitive and sensitive such thatAsand , by Lemma 1, Theorem 1.2 in [13, 42] if is D-chaotic then so is for any .
Next, we show that statement (3) is also true. Suppose that is RT-chaotic. Then, it is transitive and sensitive. By Lemmas 1 and 2, one has that is topologically transitive for any such that for any . By Theorem 1.2 in [13], is sensitive for any , which implies that is RT-chaotic for any .
Finally, by the proof of Lemma 2.1 in [13], Lemma 1 in [43], the definitions, and hypothesis, statements (5), (6), (7), and (8) are true.

Theorem 2. For a cyclic permutation map,wherethe following holds:(1) is topologically transitive if and only if is D-chaotic if and only if is RT-chaotic(2)If is topologically transitive then it is h-chaotic

Proof 3. (1)If is D-chaotic, then by definition it is RT-chaotic. If is RT-chaotic, then it is topologically transitive. Therefore, it is enough to prove that if is topologically transitive, then it is D-chaotic. Now, we let be topologically transitive. By Lemma 2, which means that is sensitive. So, is D-chaotic.(2)Suppose that is topologically transitive. By Lemma 1, is topologically transitive for any . Asby Theorem 1.2 in [13] . This implies that .

3.2. Chaos on MPE Set for a Permutation Map

For a permutation map,where

Letwhere . Let be the MPE set for .

Theorem 3. For a permutation map,wherethe following hold:(1)If is D-chaotic, then is D-chaotic(2)If is RT-chaotic, then so is (3) is h-chaotic if and only if so is (4) is LY-chaotic if and only if so is

Proof 4. As the proof of the result in Theorem 3 is similar for any , for simplicity, we only prove Theorem 3 for . First, we claim that if is topologically transitive then so is . To prove this, we let with . By the proof of Lemma 1, one can easily see that or or . By the proof of Lemma 1, if , then and . Fix . Then, . By hypothesis and the definition, there is a sequence of positive integers withFix . Then, . As is onto, there is with As Fix . Then, . As and is onto, there is with . As (1)Suppose that is D-chaotic. By the above argument, is topologically transitive. As , by hypothesis and the definition is D-chaotic.(2)Suppose that is RT-chaotic, which implies that it is topologically transitive. By the above argument, is topologically transitive. Let . Then, . By Theorem 1, there exists such that, for any open neighborhood of , there exist and an integer with This means that Similarly, the condition can be proved for any and any . By the definition, is RT-chaotic.(3)Clearly, implies that . Suppose that . Consider . It is easy to see that is trapping for and that As , which implies that .(4)Obviously, by the definition, if is LY-chaotic then so is . Suppose that is LY-chaotic. By Theorem 1, , and are LY-chaotic. Let be an uncountable Li–Yorke chaotic set for and letThen, is uncountable. Now, we will prove that is a Li–Yorke chaotic set for . For anywith , one has thatSincethere is an increasing sequence of positive integers withAs and are uniformly continuous,which mean thatConsequently, is a Li–Yorke chaotic set for , which implies that is LY-chaotic. Thus, is LY-chaotic.

3.3. The Topological Entropy and Sensitivity of a Continuous Cyclic Permutation Map

For the properties and , we refer the reader to [41]. And for the propertywe refer the reader to [44], where it is proved, in the general setting of continuous selfmaps of a compact topological space that

Moreover, the topological entropy of this kind of maps has been already computed in [45]. However, for completeness, we give Theorem 4 and its proof here.

Theorem 4. For a continuous cyclic permutation map,where for any

Proof. Asby [38] and Proposition 5.2 in [2], Theorem 4 holds.
For the sensitive properties of product maps or semiflows, we refer the reader to [43, 46]. However, for completeness, we give Theorem.5 and its proof here.

Theorem 5. For a continuous cyclic permutation map,whereand it is sensitive if and only if at least one of the following maps is sensitive:

Proof. Asby Lemma 1 in [43] and Theorem 1 in [47], Theorem 5 holds.

3.4. Discussion on Applications

It is clear that a continuous cyclic permutation map defined byfor anyin this paper is a cyclically permuted direct product (see [22, 23] and references therein). For cyclically permuted direct products, there have been many important results and applications (see [22, 23] and references therein).

When and , this type of continuous cyclic permutation map appears as certain economical model so called Cournot duopoly (see [1, 2, 10–17, 38, 42] and references therein). For many important results and applications of Cournot duopoly games, we refer the reader to [1, 2, 10–17, 38, 42] and the references therein.

From [31], we know that one can find continuous cyclic permutation maps in age-structured population models, as in [24], where it is analyzed as the Leslie model:where is a -map and each variable determines the population size of the -age class in the th period, being the initial population. To study some dynamical behavior of this kind of model is equivalent to explore the same behavior of the c.p.d.p. map (which is also a continuous cyclic permutation map), where . For study, results and applications of population models, we refer the reader to [24] and the references therein.

Because the chaotic maps have the excellent properties of unpredictability, ergodicity, and sensitivity to their parameters and initial values, they are widely used in security applications. In [35], Zhou et al. introduced a simple and effective chaotic system by a combination of two existing one-dimension (1D) chaotic maps which are called seed maps. By simulations and performance evaluations, it was shown that the proposed system can produce lots of 1D chaotic maps having larger chaotic ranges and better chaotic behaviors compared with their seed maps. To explore its applications in multimedia security, a novel image encryption algorithm is presented. By using a same set of security keys, this algorithm can generate a completely different encrypted image each time when it is used to the same original image. By experiments and security analysis it was shown that the algorithm has excellent performance in image encryption and various attacks. In [27], Hua et al. introduced a new two-dimensional Sine Logistic modulation map (2D-SLMM) which is given by the Logistic and Sine maps. When it is compared with existing chaotic maps, we can find that it has the better chaotic range, better ergodicity, hyperchaotic property, and relatively low implementation cost. Also, to investigate its applications, they proposed a chaotic magic transform (CMT) to efficiently change the image pixel positions. Combining 2D-SLMM with CMT, they further introduced a new image encryption algorithm. Simulation results and security analysis show that this algorithm can present images with low time complexity and a high security level as well as to resist various attacks. In [32], Wu et al. Noonan introduced a number of Sudoku-associated matrix element representations besides the conventional representation by matrix row-column pair. Particularly, they are representations via Sudoku matrix row-digit pair, digit-row pair, column-digit pair, digit-column pair, block-digit pair, and digit-block pair. So, one can secretly represent matrix elements by a Sudoku matrix and develop new Sudoku associated 2D parametric bijections. To show the effectiveness and randomness of thes bijections, they introduced a simple and effective Sudoku Associated Image Scrambler by 2D Sudoku-associated bijections for image scrambling without bandwidth expansion. By simulations and comparisons, it was demonstrated that the proposed method can outperform some state-of-the-art methods. In [28], Hua and Zhou gave a two-dimensional Logistic-adjusted-Sine map (2D-LASM). By performance evaluations, it was shown that this map has better ergodicity and unpredictability, and a wider chaotic range than many the existing chaotic maps. By this map, they further designed a 2D-LASM-based image encryption scheme (LAS-IES). The principle of diffusion and confusion are strictly finished, and a mechanism of adding random values to plain image is established to enhance the security level of cipher image. By simulation results and security analysis, it was shown that the LAS-IES can efficiently encrypt different kinds of images into random-like ones which have strong ability of resisting various security attacks. In [29], Hua et al. gave a sine-transform-based chaotic system (STBCS) which generates one-dimensional (1-D) chaotic maps. This chaotic system performs a sine transform to the combination of the outputs of the two existing chaotic maps (seed maps). Users have the flexibility to pick any existing 1D chaotic maps as seed maps in STBCS to generate lots of new chaotic maps. The complex chaotic behavior of STBCS is verified by using the principle of Lypunov exponent. To show the usability and effectiveness of STBCS, they presented three new chaotic maps as examples. By theoretical analysis, it was shown that these chaotic maps have complex dynamics properties and robust chaos. Performance evaluations show that they have better chaotic ranges, better complexity, and unpredictability, compared with chaotic maps generated by other methods and the corresponding seed maps. Also, to explain the simplicity of STBCS in hardware implementation, they simulated the three new chaotic maps by the field-programmable gate array (FPGA). In [30], Hua et al. presented a two-dimensional (2D) sine chaotification system (2D-SCS). Such a system can not only significantly enhance the complexity of 2D chaotic maps but also greatly extend their chaotic ranges. As examples, they applied 2D-SCS to two existing 2D chaotic maps to get two enhanced chaotic maps. By performance evaluations, it was shown that these two enhanced chaotic maps have robust chaotic behaviors in much larger chaotic ranges than the existing 2D chaotic maps. Also, a microcontroller-based experiment platform was designed to implement these enhanced chaotic maps in hardware devices. Furthermore, to discuss the application of 2D-SCS, these two enhanced chaotic maps are used to design a pseudorandom number generator. By experiment results, one can see that these enhanced chaotic maps are able to produce better random sequences than the existing 2D and several state-of-the-art one-dimensional (1D) chaotic maps. In [31], Hua et al. Zhou gave a two-dimensional (2D) modular chaotification system (2D-MCS) to improve the chaos complexity of any 2D chaotic map. Since the modular operation is a bounded transform, the chaotic maps which are improved by 2D-MCS can generate chaotic behaviors in wide parameter ranges while the existing chaotic maps cannot. Three improved chaotic maps were given as typical examples to verify the effectiveness of 2D-MCS. The chaotic properties of one example of 2D-MCS are mathematically analyzed by using Lyapunov exponent. By performance evaluations, it was shown that these improved chaotic maps have continuous and large chaotic ranges, and their outputs are distributed more uniformly than the outputs of the existing 2D chaotic maps. To explain the application of 2D-MCS, they applied the improved chaotic maps of 2D-MCS to secure communication. By the simulation results, it was shown that these improved chaotic maps exhibit better performance than a few existing and newly developed chaotic maps in terms of resisting different channel noise.

In the future, we will further discuss and explore some properties and applications of continuous cyclic permutation maps [48].

Data Availability

The data used to support the findings are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11501391), Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province (2018RZJ03), Opening Project of Bridge Non-destruction Detecting and Engineering Computing Key Laboratory of Sichuan Province (2018QZJ03), and Scientific Research Project of Sichuan University of Science and Engineering (2020RC24).

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Copyright © 2020 Risong Li and Tianxiu Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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