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.

The classical Cournot-Puu duopoly [

The aim of this paper is to investigate the parameter value

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., [

The paper is organized as follows. Section

Although topological entropy was introduced first in [

Positive topological entropy maps are chaotic in the sense of Li and Yorke (see [

The system is given by

Now, we recall that

Graphic of

We emphasize the fact that the unimodality of

Note that the dynamics of reaction functions

Next, we are going to point out an important property of our unimodal family of maps. The map

A first approach to the computation of topological entropy is given by the following fact: if

So, in view of the above results, we must concentrate our efforts in computing the topological entropy for parameter values in

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 [

The algorithm is based on several facts. The first one is that the topological entropy of the tent map family

The second one is the kneading sequence of unimodal map

If

So, the algorithm is divided in four steps.

Fix

Find the least positive integer

Compute

Find

For the practical implementation of the algorithm we use the program Mathematica, which allows us to compute the kneading sequences

Topological entropy of the map

Let us emphasize that when the accuracy is

When

It was proved in [

Figure

The picture shows the distance between the first 1000 iterates of two points for the parameter values

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

The first is a periodic orbit (recall that

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

The third is a union of periodic intervals

Moreover, the map

In order to find positive topological entropy maps for which almost all points in

Estimation of the Lyapunov exponents for

So, in view of Figure

It is well known that the existence of a periodic point of period

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 [

We show, for

Two different attractors of type A3 for the parameter value

Attractors of type A3 for

We must point out that different

Limit sets for the parameter values

Finally, if the parameter value

We analyze in detail the Cournot-Puu model introduced in [

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.