Topological Entropy of Cournot-Puu Duopoly

Abstract

The aim of this paper is to analyze a classical duopoly model introduced by Tönu Puu in 1991. For that, we compute the topological entropy of the model and characterize those parameter values with positive entropy. Although topological entropy is a measure of the dynamical complexity of the model, we will show that such complexity could not be observed.

1. Introduction

The classical Cournot-Puu duopoly [1] is a market which consists in two firms producing equivalent goods, with isoelastic demand function (in inverse form):1𝑝=𝑞₁+𝑞₂, (1.1) where 𝑞_𝑖, 𝑖=1,2, are the outputs of each firm, 𝑝 is the price, and 𝑐_𝑖, 𝑖=1,2, are the constant marginal costs. Under these assumptions, both firms maximize their profits provided that𝑞₁=𝑞₂𝑐₁−𝑞₂,𝑞₂=𝑞₁𝑐₂−𝑞₁, (1.2)(1.3) obtaining the Cournot equilibrium point (𝑞₁,𝑞₂) given by 𝑞₁=𝑐₂(𝑐₁+𝑐₂)²,𝑞₂=𝑐₁(𝑐₁+𝑐₂)². (1.4) In addition, if 𝑞_𝑖(𝑡), 𝑖=1,2, is the production of firm 𝑖 at time 𝑡, then the future production of each firm is planned according to (1.2) and (1.3), and hence, taking into account that productions cannot be negative, we obtain the (dynamical) model𝑞₁(𝑡+1)=𝑓₁(𝑞₂(𝑡))=max{𝑞₂(𝑡)𝑐₁−𝑞₂𝑞(𝑡),0},₂(𝑡+1)=𝑓₂(𝑞₁(𝑡))=max{𝑞₁(𝑡)𝑐₂−𝑞₁(𝑡),0}, (1.5) where 𝑓₁ and 𝑓₂ are called reaction functions. In [1], the stability of the Cournot point (𝑞₁,𝑞₂) was analyzed, proving that it is stable if 𝑎=𝑐₁/𝑐₂∈(3−2√2,3+2√2). In addition, a bifurcation analysis of the system was also addressed, showing numerically the existence of a period doubling bifurcation scheme.

The aim of this paper is to investigate the parameter value 𝑎=𝑐₁/𝑐₂ such that the system has a complicated dynamic behavior. To this end, we compute its topological entropy (see [2] or [3]), finding those values of positive entropy. We remark that positive entropy systems are chaotic in the sense of Li and Yorke (see [4, 5] for the definition of Li and Yorke chaos). The existence of Li-Yorke chaotic maps with zero entropy is well known (see, e.g., [6]), but we will show that they cannot exist for Cournot-Puu duopoly and so, the class of positive entropy maps agrees with the class of Li-Yorke chaotic maps.

Finally, recall that positive topological entropy and Li-Yorke chaos are topological chaos notions and hence, it is possible that “topological chaos” may not be observed in practice: for instance, when we consider a suitable unimodal map with positive entropy and an attracting periodic orbit (see e.g., [7]). We show evidences of nonobservable chaos for Cournot-Puu duopoly.

The paper is organized as follows. Section 2 is devoted to introduce the notion of topological entropy and its basic properties and to explain how we compute (approximately) it for our system. In the last section we will compare the results obtained in Section 2 with those obtained from the measure theory (physical) point of view.

2. Computing Topological Entropy

2.1. Topological Entropy

Although topological entropy was introduced first in [2], we give the Bowen definition [3] for compact metric spaces because it is more intuitive, and the connection with dynamical complexity is more clear. Let (𝑋,𝑑) be a compact metric space and let 𝑓∶𝑋→𝑋 be a continuous map. We say that a set 𝐸⊂𝑋 is (𝑛,𝜀,𝑓)-separated if for any 𝑥,𝑦∈𝐸,  𝑥≠𝑦, there exists 𝑘∈{0,1,…,𝑛−1} such that 𝑑(𝑓^𝑘(𝑥),𝑓^𝑘(𝑦))>𝜀. Denote by 𝑠(𝑛,𝜀,𝑓) the cardinality of any maximal (𝑛,𝜀,𝑓)-separated set in 𝑋. The topological entropy of 𝑓 is the non negative numberℎ(𝑓)=lim_𝜀→0limsup_𝑛→∞1𝑛log𝑠(𝑛,𝜀,𝑓). (2.1) We remark that the definitions of [2, 3] agree for continuous maps on compact metric spaces, which is our case. In addition, the topological entropy fulfills the following properties, which will be useful for us in our computations.(i)Conjugacy Invariancy [8, Chapter 7]. Let 𝑓∶𝑋→𝑋 and 𝑔∶𝑌→𝑌 be continuous maps on compact metric spaces and let 𝜋∶𝑋→𝑌 be a continuous surjective map satisfying 𝜋∘𝑓=𝑔∘𝜋. Then ℎ(𝑔)≤ℎ(𝑓). If in addition 𝜋 is a homeomorphism, 𝑔 and 𝑓 are said to be conjugated and ℎ(𝑔)=ℎ(𝑓).(ii)The Power Formula [2]. If 𝑘 is a positive integer, then ℎ(𝑓^𝑘)=𝑘ℎ(𝑓).(iii)The Product Formula [8, Chapter 7]. Let 𝑓_𝑖∶𝑋_𝑖→𝑋_𝑖, 𝑖=1,2, be continuous maps of metric compact spaces and consider the product map 𝑓₁×𝑓₂∶𝑋₁×𝑋₂→𝑋₁×𝑋₂ defined by (𝑓₁×𝑓₂)(𝑥₁,𝑥₂)=(𝑓₁(𝑥₁),𝑓₂(𝑥₂)) for all (𝑥₁,𝑥₂)∈𝑋₁×𝑋₂. Then ℎ(𝑓₁×𝑓₂)=ℎ(𝑓₁)+ℎ(𝑓₂).(iv)Commutativity Formula [9]. Let 𝑓∶𝑋→𝑌 and 𝑔∶𝑌→𝑋 be continuous maps. Then ℎ(𝑓∘𝑔)=ℎ(𝑔∘𝑓).

Positive topological entropy maps are chaotic in the sense of Li and Yorke (see [4]). Recall that 𝑓 is Li-Yorke chaotic provided that there is an uncountable set 𝑆⊂𝑋 such that for any 𝑥,𝑦∈𝑆,  𝑥≠𝑦,0=liminf_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<limsup_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦)). (2.2) The set 𝑆 is called scrambled set for 𝑓 (see [5]). However, zero topological entropy maps can be also Li-Yorke chaotic (see, e.g., [6]). We will discuss these ideas for Cournot-Puu duopoly in Section 3.

2.2. Topological Entropy of the Cournot-Puu Duopoly

The system is given by(𝑞₁(𝑡+1),𝑞₂(𝑡+1))=(𝑓₁(𝑞₂(𝑡)),𝑓₂(𝑞₁(𝑡))). (2.3) If we define 𝐹(𝑞₁,𝑞₂)=(𝑓₁(𝑞₂),𝑓₂(𝑞₁)), its second iterate is the product map:𝐹²=(𝑓₁∘𝑓₂)×(𝑓₂∘𝑓₁). (2.4) Applying the basic properties of topological entropy, we find that1ℎ(𝐹)=2ℎ(𝐹²1)=2(ℎ(𝑓₁∘𝑓₂)+ℎ(𝑓₂∘𝑓₁)), (2.5) and by the commutativity formula, ℎ(𝑓₁∘𝑓₂)=ℎ(𝑓₂∘𝑓₁). Thereforeℎ(𝐹)=ℎ(𝑓₁∘𝑓₂)=ℎ(𝑓₂∘𝑓₁). (2.6)

Now, we recall that𝑓₁(𝑞₂)=max{𝑞₂𝑐₁−𝑞₂𝑓,0},₂(𝑞₁)=max{𝑞₁𝑐₂−𝑞₁,0}, (2.7) and we assume, for instance, that 𝑐₁≥𝑐₂. If 𝜑₁(𝑞₂)=𝑞₂/𝑐₁−𝑞₂, then 𝜑₁(1/𝑐₁)=0 and 𝑞^𝑀₂=1/4𝑐₁ is the maximum of 𝜑₁ such that 𝜑₁(𝑞^𝑀₂)=𝑞^𝑀₂. Since 𝑐₁≥𝑐₂, if 𝜑₂(𝑞₁)=𝑞₁/𝑐₂−𝑞₁, then 𝑔=𝜑₂∘𝜑₁ (note that 𝑔 depends on 𝑐₁ and 𝑐₂ and we should denote it by 𝑔_𝑐1,𝑐2) has a maximum at 𝑞^𝑀₂ and 𝑞⁰₂=1/𝑐₁ is the smallest positive number such that 𝑔(𝑞⁰₂)=0. Hence, the map 𝑔∶[0,𝑞⁰₂]→[0,𝑞⁰₂] is unimodal map (it has two monotone pieces) and its maximum 𝑞^𝑀₂is called the turning point of 𝑔 (see Figure 1).

Figure 1 

Graphic of 𝑔 when 𝑐2=1 and 𝑐1=𝑎 is 5, 6.25, and 10, respectively. We see that 𝑔 has two monotone pieces.

We emphasize the fact that the unimodality of 𝑔 will play an important role in the computations of topological entropy. It is not difficult to see that the other map 𝜑₁∘𝜑₂ is not unimodal, and hence, the algorithm we are going to use to compute the topological entropy has not sense for it.

Note that the dynamics of reaction functions 𝑓₁ and 𝑓₂ is very simple. It is easy to see that 𝑓₁([0,∞))=[0,𝑞^𝑀₂] and 𝑓₁|_[0,𝑞^𝑀_2] is increasing (the same for 𝑓₂). So, the dynamics of 𝑓₁ is reduced to a repelling fixed point at 0 and an attracting fixed point 𝑞^𝑀₂. As we will see later (also in [1]), the dynamics of the joint dynamics given by 𝑓₁∘𝑓₂ and 𝑓₂∘𝑓₁ can be very complicated, in the spirit of Parrondo's paradox (see, e.g., [10, 11] or [12]).

Next, we are going to point out an important property of our unimodal family of maps. The map 𝑔=𝑔_𝑐1,𝑐2 depends on 𝑐₁ and 𝑐₂. For any 𝑥∈[0,𝑞⁰₂] we have that 𝑔_𝑐1,𝑐2(𝑥)=𝑥−𝑥𝑐₁+𝑥/𝑐₁−𝑥𝑐₂=1𝑐₂(𝑐₂𝑥−𝑐₂𝑥𝑐₁/𝑐₂+𝑐₂𝑥𝑐₁/𝑐₂−𝑐₂=1𝑥)𝑐₂𝑔_{𝑐1/𝑐2,1}(𝑐₂𝑥). (2.8) So, if 𝑎=𝑐₁/𝑐₂, 𝑔_{𝑐1/𝑐2,1}=𝑔_𝑎, and 𝜋(𝑥)=𝑐₂𝑥, we find that 𝜋∘𝑔_𝑐1,𝑐2=𝑔_𝑎∘𝜋. Hence the maps 𝑔_𝑐1,𝑐2 and 𝑔_𝑎 are topologically conjugated and their topological entropies agree. So, our problem is reduced to compute the topological entropy of the family of maps:𝑔_𝑎(𝑥)=𝑥−𝑥𝑎+𝑥𝑎−𝑥,𝑎≥1. (2.9)

A first approach to the computation of topological entropy is given by the following fact: if 𝑎≥6.25, then 𝑔_𝑎(1/4𝑎)=(2√𝑎−1)/4𝑎≥1/𝑎. Since 𝑔_𝑎 is unimodal, there are two subintervals 𝐽₁=[0,𝑥₀] and 𝐽₂=[𝑥₁,1/𝑎] such that 𝑔_𝑎(𝑥₀)=𝑔_𝑎(𝑥₁)=1/𝑎, 𝐽₁∪𝐽₂⊂𝑔_𝑎(𝐽_𝑖), for 𝑖=1,2, and [0,𝑥₀)∩(𝑥₁,1/𝑎]=∅. In other words, 𝑔_𝑎 has a 2-horseshoe and hence ℎ(𝑔_𝑎)≥log2 (see, e.g., [13, Chapter 4]. Moreover, since the topological entropy of unimodal is bounded by log2, we conclude that ℎ(𝑔_𝑎)=log2.

So, in view of the above results, we must concentrate our efforts in computing the topological entropy for parameter values in (1,6.25). The next section is devoted to that.

2.3. Practical Computation of Topological Entropy

We have to start this section by recognizing that an exact computation of the topological entropy is not possible. However, we are going to use the algorithm described in [14] to obtain good estimations of topological entropy with prescribed accuracy.

The algorithm is based on several facts. The first one is that the topological entropy of the tent map family𝑡_𝑚1(𝑥)=𝑝𝑥if0≤𝑥≤2,1𝑝−𝑝𝑥if2≤𝑥≤1 (2.10) is ℎ(𝑡_𝑝)=log𝑝, for 𝑝∈[1,2].

The second one is the kneading sequence of unimodal map 𝑓 with maximum (turning point) at 𝑐. Let 𝑓^𝑛 denote the 𝑛th iterate of 𝑓 and 𝑘(𝑓)=(𝑘₁,𝑘₂,𝑘₃,⋯) its kneading sequence given by the rule𝑘_𝑖=𝑅if𝑓^𝑖(𝑐)>𝑐,𝐶if𝑓^𝑖(𝑐)=𝑐,𝐿if𝑓^𝑖(𝑐)<𝑐. (2.11) We fix that 𝐿<𝐶<𝑅. For two different unimodal maps 𝑓₁ and 𝑓₂, we fix their kneading sequences 𝑘(𝑓₁)=(𝑘¹_𝑛) and 𝑘(𝑓₂)=(𝑘²_𝑛). We say that 𝑘(𝑓₁)≤𝑘(𝑓₂) provided that there is 𝑚∈ℕ such that 𝑘¹_𝑖=𝑘²_𝑖 for 𝑖<𝑚 and either an even number of 𝑘¹_𝑖’s are equal to 𝑅 and 𝑘¹_𝑚<𝑘²_𝑚 or an odd number of 𝑘¹_𝑖’s are equal to 𝑅 and 𝑘²_𝑚<𝑘¹_𝑚. Then one has the following.(i)If 𝑘(𝑓₁)≤𝑘(𝑓₂), then ℎ(𝑓₁)≤ℎ(𝑓₂). In addition, if 𝑘_𝑚(𝑓) denotes the first 𝑚 symbols of 𝑘(𝑓), then 𝑘_𝑚(𝑓₁)<𝑘_𝑚(𝑓₂).

So, the algorithm is divided in four steps.

Step 1. Fix 𝜀>0 (fixed accuracy) and an integer 𝑛 such that 𝛿=1/𝑛<𝜀.

Step 2. Find the least positive integer 𝑚 such that 𝑘_𝑚(𝑡_1+𝑖𝛿), 0≤𝑖≤𝑛, are distinct kneading sequences.

Step 3. Compute 𝑘_𝑚(𝑔_𝑎) for a fixed 𝑎∈(1,6.25).

Step 4. Find 𝑟 the largest integer such that 𝑘_𝑚(𝑡_1+𝑟𝛿)<𝑘_𝑚(𝑔_𝑎). Hence log(1+𝑟𝛿)≤ℎ(𝑔_𝑎)≤log(1+(𝑟+2)𝛿).

For the practical implementation of the algorithm we use the program Mathematica, which allows us to compute the kneading sequences 𝑘_𝑚(𝑡_1+𝑖𝛿) with infinite precision, that is, without round errors. Of course, the computation time increases a lot, but it must be done just one time for each fixed 𝑛. Unfortunately, it is not possible to do the computations with infinite precision for our family 𝑔_𝑎. Figure 2 shows the topological entropy estimations of 𝑔_𝑎.

(a)

(b)

Figure 2 

Topological entropy of the map 𝑔𝑎. (a) We can find the entropy for 𝑎∈[6.18,6.25] with accuracy of 10−3. (b) We can see a zoom close to 𝑎=6.186, when the topological entropy becomes to be positive. The accuracy for the right picture is 10−5.

Let us emphasize that when the accuracy is 10⁻³ (i.e., 𝑛=1000), then the length of the kneading sequences of the tent map is 𝑚=896. For accuracy of 10⁻⁶ the length 𝑚=786432 distinguishes between the kneading sequence of tent maps with 𝑝=1+10⁻⁶ and 𝑝=1+2⋅10⁻⁶. So, with accuracy of 10⁻⁶, we find that the first parameter 𝑎 with positive entropy (greater than log(1.000001)) is 6.186688432716827. Finally, note that positive topological entropy is only possible when the constant marginal costs are very different; that is, only “different” firms can produce chaos.

3. On Chaos and Further Discussions

When 𝑔_𝑎 has positive topological entropy, it is chaotic in the sense of Li and Yorke. The Cournot-Puu map 𝐹 is Li-Yorke chaotic if and only if both maps 𝑓₁∘𝑓₂ and 𝑓₂∘𝑓₁ are also Li-Yorke chaotic (see [15]). Although the existence of Li-Yorke chaotic maps with zero topological entropy is known (see, e.g. [6]), we will show that such maps cannot exist in the Cournot-Puu model. The argument is as follows.

It was proved in [16] that zero topological entropy interval maps are Li-Yorke chaotic if they have wandering intervals (recall that a subinterval 𝐽⊂[𝑎,𝑏] is wandering for a continuous map 𝑓∶[𝑎,𝑏]⊂ℝ→[𝑎,𝑏] if for any 𝑛,𝑚∈ℕ, 𝑓^𝑛(𝐽)∩𝑓^𝑚(𝐽)=∅). It was proved in [17] that 𝐶^∞ maps with nonflat turning points do not have wandering intervals (A turning point 𝑐 is nonflat if some derivative of 𝑓 at 𝑐 does not vanish.). Of course, our map 𝑔_𝑎, 𝑎∈(1,6.25), has no finite derivative at 0, but the dynamics of 𝑔_𝑎 is concentrated in the invariant (by 𝑔_𝑎) interval [𝑔²_𝑎(1/4𝑎),𝑔_𝑎(1/4𝑎)], in which the map is 𝐶^∞. The second derivative at the turning point is given by𝑔_𝑎^′′(1√4𝑎)=2𝑎(1−𝑎), (3.1) which is different from zero for all 𝑎∈(1,6.25). As a consequence, the turning point of 𝑔_𝑎 is nonflat and therefore 𝑔_𝑎 has not wandering intervals. Hence, in the case of the Cournot-Puu model, the class of positive topological entropy maps agrees with the class of Li-Yorke chaotic maps.

Figure 3 shows the distance ‖𝐹^𝑛(𝑥₁,𝑦₁)−𝐹^𝑛(𝑥₂,𝑦₂)‖ for different parameter values. Recall that the existence of two points holding the Li–Yorke chaos conditions (called a Li-Yorke pair) is enough to guarantee the chaoticity of the map (see [15]).

(a)

(b)

Figure 3 

The picture shows the distance between the first 1000 iterates of two points for the parameter values 𝑎=6.24 (a) and 𝑎=6.25 (b).

Topological entropy and Li-Yorke chaos are purely topological notions. We must point out that such topological chaos could not be observed, say on a computer simulation. More precisely, we are going to show that there are parameter values for which the Cournot-Puu duopoly displays positive topological entropy while almost any point (in the sense of the Lebesgue measure) is attracted by a periodic orbit. In other words, the probability of a single orbit to be attracted by a periodic point is one. This leads us to the philosophical question on whether chaos exists when it cannot be observed.

Fix 𝑥∈[0,1/𝑎] and denote the set of limit points of the sequence (𝑔^𝑛_𝑎(𝑥)) by 𝜔(𝑥,𝑔_𝑎), which will be called the 𝜔-limit set of 𝑥 under 𝑔_𝑎. Recall that a metric attractor is a subset 𝐴⊂[0,1/𝑎] such that 𝑔_𝑎(𝐴)⊆𝐴, ℬ(𝐴)={𝑥∶𝜔(𝑥,𝑔_𝑎)⊂𝐴} has positive Lebesgue measure, and there is no proper subset 𝐴⫅^′_/𝑞𝐴 with the same properties. By [18], the regularity properties of 𝑔_𝑎 imply that there are three possibilities for its metric attractors.(A1)The first is a periodic orbit (recall that 𝑥 is periodic if 𝑔^𝑛_𝑎(𝑥)=𝑥 for some 𝑛∈ℕ).(A2)The second is a solenoidal attractor, which is basically a Cantor set in which the dynamics is quasiperiodic. More precisely, the dynamics on the attractor is conjugated to a minimal translation, in which each orbit is dense on the attractor. The dynamics of 𝑔_𝑎 restricted to the attractor is simple; neither positive topological entropy nor Li-Yorke chaos can be obtained.(A3)The third is a union of periodic intervals 𝐽₁,…,𝐽_𝑘, such that 𝑔^𝑘_𝑎(𝐽_𝑖)=𝐽_𝑖 and 𝑔^𝑘_𝑎(𝐽_𝑖)=𝐽_𝑗, 1≤𝑖<𝑗≤𝑘, and such that 𝑔^𝑘_𝑎 is topologically mixing. Topologically mixing property implies the existence of dense orbits on each periodic interval (under the iteration of 𝑔^𝑘_𝑎).

Moreover, the map 𝑔_𝑎 can have at most one attractor of type A2 and A3 for any 𝑎∈(1,6.25).

In order to find positive topological entropy maps for which almost all points in [0,1/𝑎] are attracted by a periodic orbit, we have to exclude the possibility of existence of attractors of types A2 and A3. Since an attracting periodic orbit {𝑥₀,𝑔_𝑎(𝑥₀),…,𝑔_𝑎^𝑘−1(𝑥₀)} holds that𝑑=|𝑔^′_𝑎(𝑥₀)𝑔^′_𝑎(𝑔_𝑎(𝑥₀))⋯𝑔^′_𝑎(𝑔_𝑎^𝑘−1(𝑥₀))|<1, (3.2) we estimate the Lyapunov exponent𝜆(𝑥,𝑎)=limsup_𝑛→∞1𝑛log|𝑔^′_𝑎(𝑔^𝑛_𝑎(𝑥))| (3.3) for 𝑥=𝑔¹⁰⁶_𝑎(1/4𝑎). Since the metric attractor either contains the turning point or is an attracting periodic orbit (see [18]), we have that those values of 𝑎 for which 𝜆(𝑥,𝑎)<0 are good candidates to have only periodic attractors. In Figure 4 we show the estimations of the Lyapunov exponent with 𝑛=1000.

(a)

(b)

Figure 4 

Estimation of the Lyapunov exponents for 𝑎=6.186+𝑘/1000, 0≤𝑘<640 (a) and 6.2115+𝑘/100000, 0≤𝑘≤400 (b).

So, in view of Figure 4, for instance, we consider 𝑎₀=6.21236. We find that 𝑥₀≅0.160363 is a period 6 such that𝑑=|Π⁵_𝑘=0𝑔^′_𝑎0(𝑔^𝑘_𝑎0(0.160363))|≅0.003, (3.4) and hence we have a periodic attractor, while the topological entropy of 𝑔_𝑎 is positive. So, if almost all orbits are attracted by a periodic point, Li-Yorke pairs cannot be observed because for any 𝑥,𝑦∈(0,1/𝑎) we have that 𝑥 and either 𝑦 are proximal, that is,lim_𝑛→∞|𝑔^𝑛_𝑎(𝑥)−𝑔^𝑛_𝑎(𝑦)|=0, (3.5) or distalliminf_𝑛→∞|𝑔^𝑛_𝑎(𝑥)−𝑔^𝑛_𝑎(𝑦)|>0. (3.6) Note that this periodic attractor of 𝑔_𝑎 generates six different periodic attractors, with period 12, of the two-dimensional Cournot-Puu map. The spatial distribution of these periodic attractors was characterized in [19]. In [20], the existence of different attractor for two dimensional maps was also pointed out.

It is well known that the existence of a periodic point of period 6 of 𝑔_𝑎 implies that ℎ(𝑔_𝑎)>0 (see, e.g., [13, Chapter 4]). So, for the parameter value 𝑎=6.21236 the reader may suspect the existence of chaos for 𝑔_𝑎. The next example is even more paradoxical. For the 𝑎=6.2085 there is an attracting periodic point of period 64 and, since periodic points whose periods are power of 2 do not guarantee that 𝑔_𝑎 had positive topological entropy, the existence of the above periodic orbit of period 64 does not offer any implication of topological chaos. Since almost all orbits are attracted to this periodic orbit (at least numerical experiments show it), the topological chaos could not be detected by evaluating the orbits.

In the case of positive Lyapunov exponent, we can address the existence of periodic rectangles as metric attractors, which follow the spatial distributions characterized in [21]. We call them attractors of type A3, because they are generated by attractors of type A3 of 𝑔_𝑎. For instance, for 𝑎=6.189, we find two different metric attractors given by four periodic rectangles. For 𝑎=6.24 we have a unique attractor given by four periodic topologically mixing rectangles. In the limit case 𝑎=6.25 we find a topologically mixing map on a rectangle. Figures 5, 6, and 7 show these attractors. It is unclear for us whether solenoidal type attractors exist in this model.

Figure 5 

We show, for 𝑎=6.1872, an attractor of type A3. It consists of 16 periodic rectangles, each of them being transitive under the iteration of the two dimensional map 𝐹𝑎16.

(a)

(b)

Figure 6 

Two different attractors of type A3 for the parameter value 𝑎=6.189.

(a)

(b)

Figure 7 

Attractors of type A3 for 𝑎=6.24 (a). For the limit case 𝑎=6.25 we receive a dense orbit on the rectangle, which is the attractor of the system (b).

We must point out that different 𝜔-limit sets of the system can be obtained by computer simulations, even when they are not attractors of the system. For instance, for parameters values 𝑎=6.24 and 𝑎=6.25, we can obtain in Figure 8 the following limit sets of the point (0.001,𝑓₂(0.001)), which is in the so-called MPE set of the system given by the points in the set {(𝑥,𝑓₂(𝑥))∶𝑥≥0}∪{(𝑓₁(𝑥),𝑥)∶𝑥≥0}. In general, the 𝜔-limit sets for points in the sets ℳ_𝑘={(𝑥,𝑓₂((𝑓₁∘𝑓₂)^𝑘(𝑥)))∶𝑥≥0}∪{(𝑓₁((𝑓₂∘𝑓₁)^𝑘(𝑥)),𝑥)∶𝑥≥0} (3.7) for 𝑘=1,2,… are also one-dimensional. Figure 9 shows some examples. These 𝜔-limit sets are obtained when the associated one dimensional orbits of the model satisfy some synchronization properties.

(a)

(b)

Figure 8 

Limit sets for the parameter values 6.24 (a) and 6.25 (b). They can be observed although they are not properly an attractor of the system.

(a)

(b)

Figure 9 

𝜔-limit sets for parameter value 𝑎=6.24 and 𝑘=6 (a) and 𝑘=16 (b).

Finally, if the parameter value 𝑎 is greater than 6.25, although the entropy is positive, we can see that the orbit of almost any point is eventually the fixed point 0. The reason for that is the existence of an interval 𝐽⊂[0,1/𝑎] such that 𝑔_𝑎(𝑥)>1/𝑎 for any 𝑥∈𝐽, and therefore 𝑔²_𝑎(𝑥)=0. We say that this map has a “hole” and all the interval [0,1/𝑎] except for a Cantor set goes eventually to 0, and chaos cannot be detected. We refer the reader to [22, 23] for more information about the dynamics of unimodal maps with holes.

4. Conclusion

We analyze in detail the Cournot-Puu model introduced in [1]. We use very recent and powerful mathematical results to characterize the attractors of the model and to compute its topological entropy. Although the existence of topological chaos is proved when the difference between the marginal costs of both firms is big enough, we check that it may not be physically observed. The following questions remain open for this model. On one hand, it would be useful to get another algorithm for computing the topological entropy in order to improve the accuracy of the computations. On the other hand, it is unclear how many attractors in the system. Note that the map 𝑔_𝑎 might have more than one attractor because its Schwartzian derivative need not be negative (recall that unimodal maps with negative Schwartzian derivative have only one attractor [7]). It is also an open question whether the existence of solenoidal attractors is possible, and how they are distributed in the square in the case of the two-dimensional model.

Acknowledgments

Jose S. Cánovas has been partially supported by the Grants MTM2008–03679/MTM from Ministerio de Ciencia e Innovación (Spain) and FEDER (Fondo Europeo de Desarrollo regional) and 08667/PI/08 from Programa de Generación de Conocimiento Científico de Excelencia de la Fundación Séneca, Agencia de Ciencia y Tecnología de Ia Comunidad Autónoma de la Región de Murcia (II PCTRM 2007–10). This paper was written while Jose S. Cánovas was enjoying the program “Intensificación de Ia actividad investigadora” at Universidad PoIitécnica de Cartagena.