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 ,