Chaos in Duopoly Games via Furstenberg Family Couple

Zhao, Yu; Li, Risong

doi:https://doi.org/10.1155/2019/5484629

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

Research Article | Open Access

Volume 2019 | Article ID 5484629 | https://doi.org/10.1155/2019/5484629

Chaos in Duopoly Games via Furstenberg Family Couple

Yu Zhao¹and Risong Li¹

Academic Editor: Marcin Mrugalski

Received04 Jul 2019

Accepted25 Oct 2019

Published23 Nov 2019

Abstract

Assume that and are two given closed subintervals of and that and are continuous maps. Let be a Cournot map over the space . In this paper, we study -chaos (resp. strong -chaos) of such a Cournot map. We will show that the following are true: (1) is -chaotic (resp. strong -chaotic) if and only if is -chaotic (resp. strong -chaotic) if and only if is -chaotic (resp. strong -chaotic). (2) is -chaotic (resp. strong -chaotic) if and only if is -chaotic (resp. strong -chaotic). (3) is -chaotic (resp. strong -chaotic) if and only if so is . MR(2000) Subject Classification: Primary 37D45, 54H20, and 37B40 and Secondary 26A18 and 28D20.

1. Introduction

Let and be closed subintervals of , and let and be continuous. In the whole paper, is defined by for any . Such a map has been investigated to give a mathematical analysis of Cournot duopoly (see [1]). Probably the first notion of chaos in a mathematically rigorous way was posed by Li and Yorke [2]. Since then, a lot of different notions of chaos have been posed. Akin and Kolyada gave the concept of Li–Yorke sensitivity for the first time [3]. They also gave the concept of spatiotemporal chaos. Schweizer and Smítal gave the concept of distributional chaos [4]. We know that distributional chaos is equivalent to positive topological entropy and some other chaotic properties for some particular spaces (see [4, 5]), and that this equivalence relationship will become invalid for some higher dimensional spaces [6] and some zero-dimensional spaces [7]. In [8], Wang et al. gave the definition of distributional chaos with respect to a sequence and got that such chaos is equivalent to Li–Yorke chaos for continuous maps over a closed subinterval. Over the past few decades, people have been paying very close attention to the chaotic properties of Cournot maps (see [1,9–13]). From [1, 12] one can see that there exist Markov perfect equilibria processes. That is, two fixed players move alternatively and ensure that any of them chooses the best reply to the previous action of another player. Put , , and . Obviously, . The set is said to be a MPE set for (see [9]). Moreover, in [9], the authors studied several kinds of chaos for Cournot maps and obtained that for any definition they considered in [9], and it does not satisfy the condition that is chaotic if and only if so is . It is well known that some chaotic properties of Cournot maps have been explored (see [1,12–17]). Recently, Lu and Zhu further studied some chaotic properties of Cournot maps and showed that some chaotic properties of , and are same. In this paper, it is shown that for any Cournot map over the product space , the following properties are hold:(1) is -chaotic (resp. strong -chaotic) if and only if is -chaotic (resp. strong -chaotic) if and only if is -chaotic (resp. strong -chaotic).(2) is -chaotic (resp. strong -chaotic) if and only if is -chaotic (resp. strong -chaotic)(3) is -chaotic (resp. strong -chaotic) if and only if so is

2. Preliminaries

Let be a compact metric space. A dynamic system means that f is a continuous self-map over the space H.

Let be a map on the space . The map f is chaotic in the sense of Li–Yorke if there is an uncountable set satisfying that for any with :

This uncountable set is called a scrambled set of f.

An important generalization of Li–Yorke chaos is distributional chaos, which is given in 1994 by Puu and Sushko [1].

Let be a metric space and be continuous. For any , the upper (lower) distribution function () deduced by and f is defined bywhere is the characteristic function of the set . The map f is distributional chaotic if there is an uncountable subset satisfying that for any , , and such that . This uncountable subset is called a distributionally scrambled set of f. And this point pair which satisfies the above two conditions is called a distributionally scrambled pair of f.

In 1997, Furstenberg family is introduced by Akin [18]. Then, Xiong and Tan defined -chaos and described chaos via Furstenberg family couple. Also, they obtained some sufficient conditions of -chaos (see [19]).

Let , , and be the collection of all subsets of . A collection is called a Furstenberg family (see [19]) if it satisfies that if and then . A family is said to be proper if it is a proper subset of (see [19]). In the whole paper, we suppose that all Furstenberg families are proper. Clearly, a family is proper if and only if and (see [19]).

For any Furstenberg families and and any map , is called a -scrambled set of f (see [19]), if , the following two conditions are satisfied:(1), (2),

This pair is called a -scrambled pair of f. The map f is said to be -chaotic if there is an uncountable -scrambled set of f. When , the map f is said to be -chaotic and the pair is a -scrambled pair. The map f is said to be strong -chaotic if one can find satisfying that for any , , and .

Similarly, one can give the concept of strong -chaos.

Let . The upper density and the lower density of G are defined bywhere denotes the cardinality of the set G.

Let denotes the set of all infinite subsets of . For arbitrary , put . Then, a pair is a -scrambled pair if and only if it is a Li–Yorke scrambled pair (see [19]). A pair is a -scrambled pair if and only if it is a distributionally scrambled pair (see [19]). Hence, -chaos is Li–Yorke chaos, and -chaos is distributional chaos.

For and , let and . A Furstenberg family is said to be translation-invariant if for any and any and . It is easily seen that is a proper and translation-invariant family (see [19]).

Clearly, for any , is a translation-invariant Furstenberg family and = (see [19]).

3. Main Results

Theorem 1. Let the product metric ξ on the product space be defined by and the product map of and be defined by for any and any , where are compact intervals, and let be a Cournot map. If and are two Furstenberg families such that is translation-invariant, then is -chaotic if and only if so is .

Proof. Suppose that is -chaotic. By the definition, there is an uncountable -scrambled set of . By the definition, for any given and any with one has thatAs is uniformly continuous, for any there is such that and imply that . So, ifthenConsequently, byBy the definition, for any with there is such thatAs is uniformly continuous, for the above there is such that and imply that . So, ifthenAs is translation-invariant, byThis means thatThus, Theorem 1 is true.

Theorem 2. Let the product metric ξ on the product space be defined by and the product map of and be defined by for any and any , where are compact intervals, and let be a Cournot map. If and are two Furstenberg families such that is translation-invariant, then is strong -chaotic if and only if so is .

Proof. Suppose that is strong -chaotic. By the definition, there is an uncountable strong -scrambled set of . By the definition, for any given and any with one has thatAs is uniformly continuous, for any there is such that and imply that . So, ifthenConsequently, byBy the definition, for any with there is satisfying that for any with , one has thatAs is uniformly continuous, for the above there is such that and imply that . So, ifthenAs is translation-invariant, byThis means thatThus, Theorem 2 is true.

Corollary 1. Let be a Cournot map on the product space . Then, for any , is -chaotic (resp. strong -chaotic) if and only if so is .

Proof. As is a translation-invariant Furstenberg family for any , by Theorems 1 and 2 one can see that Corollary 1 holds.

Theorem 3. Let be a Cournot map on the product space . If and are two Furstenberg families such that is translation-invariant and satisfy that for any and any , there is satisfying that , and that for any and any ,then is -chaotic if and only if so is .

Proof. We assume that is -chaotic.

Claim 1. is -chaotic.

The Proof of Claim 1. Assume that is a -scrambled set of the system . As and are uniformly continuous, for any there is satisfying that and imply and . By the hypothesis and the definition, for any with , one has thatAs satisfies that for any , there is satisfying that , by the definition there is satisfying thatBy the above argument, one has thatThat is,So,As is translation-invariant,By the hypothesis and the definition, for any given with there is satisfying thatAs satisfies that for any , there is satisfying that , by the definition there is satisfying thatAs and are uniformly continuous, for the above , there is satisfying that and imply and . Clearly,which means thatThus, Claim 1 holds.
As , by hypothesis, Claim 1, the definition of -chaos, and Theorem 1 and its proof, one can easily verify that and are -chaotic.
Assume that is -chaotic. By the definition, there is an uncountable subset which is -scrambled set of . By the proof of Theorem 1, is an uncountable -scrambled set of . Set . Then, A is uncountable. By the above argument, the definition of -chaos and the proof of Theorem 1, it is easy to prove that A is a -scrambled set of .
Now, we assume that is -chaotic.

Claim 2. is -chaotic.

The Proof of Claim 2. By the hypothesis and the definitions, is -chaotic. Assume that D is a -scrambled set of the system . As is uniformly continuous for any , for any there is satisfying that and imply for any . By hypothesis and the definition, for any with , one has thatSo, for any and any we have thatAs satisfies that for any ,by the definition we haveClearly,This means thatBy the hypothesis and the definition, for any given with there is satisfying thatAs is uniformly continuous for any , for the above there is satisfying that and imply for any . So, for any and any we have thatAs is translation-invariant, As satisfies that for any ,Clearly,which means thatThus, Claim 2 holds.
Consequently, Theorem 3 is true.

Theorem 4. Let be a Cournot map on the product space . If and are two Furstenberg families such that is translation-invariant and satisfy that for any and any , there is satisfying that , and that for any and any ,then is strong -chaotic if and only if so is .

Proof. We assume that is strong -chaotic.

Claim 3. is strong -chaotic.

The Proof of Claim 3. Assume that is a strong -scrambled set of the system . As and are uniformly continuous, for any there is satisfying that and imply and . By hypothesis and the definition, for any with , one has thatAs satisfies that for any ,by the definition there is satisfying thatBy the above argument, one has thatThat is,So,As is translation-invariant,By the hypothesis and the definition, there is such that for any with ,As satisfies that for any , there is satisfying that , by the definition there is satisfying thatAs and are uniformly continuous, for the above there is satisfying that and imply and . Clearly,which means thatThus, Claim 3 holds.
As , by hypothesis, Claim 3, the definition of strong -chaos, and Theorem 2 and its proof, one can easily verify that and are strong -chaotic.
Assume that is strong -chaotic. By the definition, there is an uncountable subset which is strong -scrambled set of . By the proof of Theorem 2, is an uncountable and strong -scrambled set of . Set . Then, A is uncountable. By the above argument, the definition of strong -chaos and the proof of Theorem 2, it is easy to prove that A is a strong -scrambled set of .
Now, we assume that is strong -chaotic.

Claim 4. is strong -chaotic.

The Proof of Claim 4. By the hypothesis and the definitions, is strong -chaotic. Assume that D is a strong -scrambled set of the system . As is uniformly continuous for any , for any there is satisfying that and imply for any . By the hypothesis and the definition, for any with , one has thatSo, for any and any , we have thatAs satisfies that for any ,by the definition we haveClearly,This means thatBy the hypothesis and the definition, there is satisfying that for any with ,As is uniformly continuous for any , for the above there is satisfying that and imply for any . So, for any and any , we have thatAs is translation-invariant, As satisfies that for any ,Clearly,which means thatThus, Claim 4 holds.
Consequently, Theorem 4 is true.

Corollary 2. Let be a Cournot map on the product space . Then, for any , is -chaotic (resp. strong -chaotic) if and only if so is .

Proof. We have the following two claims.

Claim 5. For any , satisfies that for any , there is such that .

The Proof of Claim 5. It is clear that and that if , then there is satisfying that . Assume that there is such that does not have the property . By this assumption and the definition, there is such that for any , . Choose for any . As for any , by the definition there is an integer such that for any and any integer , . This implies thatLet and write , where is the integral part of and . By the definition, implies for any . Obviously, if and then for any . So,whereThis implies thatSet . AsBy ,This is a contraction. Consequently, Claim 5 holds.

Claim 6. For any , satisfies that for any :

The Proof of Claim 6. It is obvious that , and that if , then . Assume that there is such that does not have the property . By this assumption and the definition, there is such that . Choose . As , by the definition there is an integer such that for any integer , . Let and write , where is the integral part of and . By the definition, implies for any . Obviously, if and , then for any . So,This implies thatConsequently,This is a contraction. Consequently, Claim 5 is true.
From the above two claims we know that satisfies the conditions of Theorems 3 and 4 for any . Thus, by these two theorems one can see that Corollary 2 holds.

Theorem 5. Let be a Cournot map on the product space . If and are two Furstenberg families such that is translation-invariant, then is -chaotic (resp. strong -chaotic) if and only if so is .

Proof. The proof is similar to those of Theorems 3 and 4 and is omitted.

Corollary 3. Let be a Cournot map on the product space . Then, for any , is -chaotic (resp. strong -chaotic) if and only if so is .

Proof. By Theorem 5 and the proof of Corollary 2 one can easily see that Corollary 3 holds.

Theorem 6. Let be a Cournot map on the product space . If and are two Furstenberg families such that is translation-invariant, then is -chaotic (resp. strong -chaotic) if and only if so is .

Proof. By hypothesis, the definitions of , , and -chaos (resp. strong -chaos) and Theorems 3 and 4, it is easily seen that is -chaotic (resp. strong -chaotic) if and only if so is .

Corollary 4. Let be a Cournot map on the product space . Then, for any , is -chaotic (resp. strong -chaotic) if and only if so is .

Proof. By Theorem 6 and the proof of Corollary 2 one can easily see that Corollary 4 holds.

Data Availability

The data used to support the findings of this study 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 project was supported by the Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province (No. 2018RZJ03) and the Opening Project of Bridge Nondestruction Detecting and Engineering Computing Key Laboratory of Sichuan Province (2018QZJ03).

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Copyright

Copyright © 2019 Yu Zhao and Risong Li. 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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