Describing the Dynamics and Complexity of Matsumoto-Nonaka ’ s Duopoly Model

and Applied Analysis 3


Introduction: The Model and Our Aims
In [1] a two-market model consisting in two firms, which produce differentiated goods is introduced.The first firm produces good  in the first market, and the second one produces good  in the second market.It is assumed that externalities of different signs exist.An externality occurs when the actions of firms directly affect the production capabilities of other firms other than through the price mechanism of the market.In this case, the positive externality comes from the market demand; that is, the sales possibilities of one firm are positively influenced by the production of the other firm.The negative externality is due to the cost functions of each firm, since the cost depends not only on its own production but also on the other firm's production.
In this frame, we are ready to introduce the equations of the model.Although in [1] a more general model is introduced, we are interested in the following particular case.Inverse demand functions are given by  1 (, ) = ( − 1) 2 where  1 and  2 are the market prices of goods  and , respectively, and  ∈ [1,2] ⊂ R and  ∈ [0, 2] ⊂ R. Each firm decides future production depending on the other firm's choice and production externalities.Cost functions are given by and thus, the profit function of each firm is given by ( It is assumed that each firm tends to maximize its profit.In order to get it, the firms can choose their production levels which would affect the other firm.The first firm maximizes the profit with respect to , and the same occurs for the second firm respect to , that is, Hence, by solving (4), and under naive future expectations, the reaction functions of the firms are given by   () = ( −  + 1) 2 ,   () = ( − 1) 2 . ( This is a static situation, but we are interested in the dynamic interactions between the two firms along the time.The function  , (, ) = (  () ,   ()) = (( −  + 1) 2 , ( − 1) 2 ) ( is the reaction function for the outcome (, ).The iterations are given by  +1 =   (  ) = (  −  + 1) 2 ,  +1 =   (  ) = (  − 1) 2 ; that is, where , is not chosen in an arbitrary way.
Observe that the reaction map  , is a map from or, in other words, and therefore, one might expect that the dynamics of the whole system can be derived from the one-dimensional maps  , : [0, 1] → [0, 1] and  , : [0, 1] → [0, 1] given by Our aims are, on one hand, to give an analytical proof of the existence of chaos in the model.This proof will be given by computations of topological entropy of the system with prescribed accuracy.As positive topological entropy will imply the existence of chaos in the sense of Li and Yorke, we will be able to give a proof of existence of chaos for a wide range of parameter values.On the other hand, we will be able to describe the nature of attractors of the system, and we will go beyond to the seminal work [1] proving the existence of absolutely continuous ergodic measures.The existence of such measures will allow us to obtain some consequences from both dynamic and economic points of view.Finally, we must emphasize that our approach is somewhat different from that made in [2].Moreover, our scheme can be adapted to analyze the dynamics of models given by maps with the form of  , .
The paper is organized as follows.In Section 2 we study the complexity of the model.For that, we compute the topological entropy using different algorithms depending on the number of pieces of monotonicity of the functions involved in the model.As an approach to the attractors, their number is studied in Section 3. On one hand, the fact that the Schwarzian derivative of the function is negative allows us to know the possibilities of the metric attractors that can appear.On the other hand, a real computation of the attractors allows us to realize that topological chaos need not be physically observed, with the Lyapunov exponents being the key for analyzing these phenomena.The Lyapunov exponent gives us the key to study that situation.Invariant measures and average profits are considered in Section 4. Finally, Section 5 is devoted to conclusions.

Computing Topological Entropy
The goal of this section is to compute the topological entropy of the system.Topological entropy was introduced in [  a continuous piecewise monotone function, the Misiurewicz-Szlenk's theorem, [4], gives us the following characterization of the topological entropy ℎ().Theorem 1.Let : [0, 1] → [0, 1] be a piecewise continuous monotone functions and () the number of pieces of monotonicity.Then, Although our system is two-dimensional, the computation of topological entropy can be reduced to a onedimensional problem.For that, we use the power formula [3], and the commutativity formula [5], to obtain Next, we must recognize that Misiurewicz-Szlenk formula is not useful for computing topological entropy of our model, so we are forced to use numerical algorithms which are based on the number of monotone pieces of  , and  , .Therefore, we will make a precise description of monotonicity regions for these maps.

Monotonicity
Regions of  , and  , .We study the shape of the composition map, where shape means in this    case the number of pieces of monotonicity that each map has depending on the values of the parameters  and .This will be used in the next section to compute the topological entropy.
The map , in terms of monotonicity, has the following structure in the parametric space (see Figure 1). where see example for  = 1.5 and  = 0.3 in Figure 2(a).(ii)  , is a unimodal map with a minimum for  = where (iii)  , is a bimodal map with a minimum for  = (1/ )(1 − √1 − 1/) and a maximum for where see example for  = 1.5 and and a maximum for  = 1/ if (, ) ∈  3 , where see example for  = 1.5 and  = 1.8 in Figure 2(d).
(i)  , is a unimodal map with a maximum (for where see example for  = 1.5 and  = 0.6 in Figure 3(a).(ii)  , is a bimodal map with a maximum for  = 1−1/ and a minimum for where see example for  = 1.9 and  = 1.9 in Figure 3(c).

Practical Computation of Topological Entropy.
Algorithms for computing the topological entropy for the special case of unimodal maps are given in [6,7].An algorithm for bimodal maps (with three pieces of monotonicity) is described in [8], and an algorithm for a particular class of piecewise continuous maps with four pieces of monotonicity is described in [9].These three types of algorithms cover all the cases involved in the dynamical system that we are considering for the composition maps  , and  , .The results obtained for the model can be observed in Figure 4.   Positive topological entropy implies chaos in the sense of Blanchard et al. [10], and therefore, the above computations of topological entropy are an analytical proof of the existence of chaos in the model.Even more, we can prove that positive topological entropy is equivalent to chaos in the sense of Li-Yorke.For that, we just need to notice that when the topological entropy is equal to zero, the map is not chaotic because the relative extreme points of the composition maps   ∘   and   ∘   are nonflat since the second derivative in these points is nonzero, (remember that a critical point is nonflat if some higher derivative is nonzero).Then, by [11, Theorem A],  ∞ maps with nonflat turning points have no wandering intervals (i.e., for a continuous interval map , an interval  is called wandering if , (), () 2 , . . .are disjoint and no point  ∈  is asymptotically periodic), and by [12, Lemma 2.7], a map with zero topological entropy is chaotic if and only if it has wandering intervals.Hence, our model is not chaotic when the topological entropy is zero.
Therefore, we have two regions the nonchaotic one where every trajectory has the property that for any  > 0 there is a periodic trajectory which is  close to it [13].In practice, one might check that a computer simulation shows the convergence to a periodic orbit when parameters are in the nonchaotic region.For the second region one could expect a similar situation, but as we will show in the next section, the topological chaos we have shown to exist, could not be observed.For instance, for  = 1.8 and  = 1.85, the topological entropy of   ∘   is positive, but the time series of several orbits reveal a periodic motion (see Figure 7).

Chaos, Attractors, and Schwarzian Derivative
We start this section with some useful definitions.Let  :  →  be a map.A probabilistic measure  is said to be invariant for  if () = ( −1 ()) for any Borel set  ⊂ .In addition, the measure is ergodic if the equality () =  implies that () = 0 or 1. Denote by M(, ) and E(, ) the set of invariant and ergodic measures of , respectively.Given  ∈ , define its -limit set (, ) as the set of limits points of its orbit.Recall that a metric attractor is a subset  ⊂ [0, 1] such that () ⊆ , () = { : (, ) ⊂ } has positive Lebesgue measure, and there is no proper subset   ⫋  with the same properties.() is called the basin of the attractor.
By [14, Theorem 4.1], for a multimodal map  :  →  without wandering intervals, there are three possibilities for its metric attractors.(A1) A periodic orbit (recall that  is periodic if   () =  for some  ∈ N).
(A2) 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  restricted to the attractor is simple; neither positive topological entropy nor Li-Yorke chaos can be obtained.Its dynamics is often known as quasiperiodic.(A3) A union of periodic intervals,  1 , . . .,   , such that   (  ) =   and   (  ) ̸ =   , 1 ≤  <  ≤ , and such We make 200000 iterations and draw the last 100000 in the  axis.We can see that for these parameter values the function is chaotic, but it could happen that the chaos is not physically observable.
that   is topologically mixing.Topologically mixing property implies the existence of dense orbits on each periodic interval (under the iteration of   ).
Moreover, if  has an attractor of types (A2) and (A3), then it must contain the orbit of a turning point, and therefore its number is bounded by the turning points.In the case of our model, we can strengthen the above result by noticing that  , and  , have negative Schwarzian derivative.Namely, the Schwarzian derivative [15][16][17] at a point  is given by Since ( ∘ )() = (()) ⋅ (  ()) 2 + () and the maps   and   have negative Schwarzian derivative the same occurs to the composition maps  , and  , .Since attractors of type (A1) must also attract the orbit of a turning point when the map has negative Schwarzian derivative, we have that the number of attractors of  , and  , is bounded to at most 3.Even more, when  , and  , have three monotone pieces, then their value in two turning points agree, which implies that in fact, the number of attractors of  , and  , can be at most 2. The following result shows that the attractors of  , and  , are deeply connected.
Proof.As usual, denote by  the one-dimensional Lebesgue measure.Let  ⊂  with () > 0 such that for any  ∈  we have (,  ∘ ) = (,  ∘ ).Since  is  1 , we have that ( −1 ()) > 0. Now, let  ∈  −1 (), and fix  ∈  such that  = ().For any  ∈ N we have from which we conclude that ((),  ∘ ) = ((,  ∘ )) is an attractor of  ∘ .The attractors of the map  , can be obtained by combining the attractors of   ∘   and   ∘   .Basically, we must use the inclusion [18, Theorem 2] and, in view of Proposition 2, check whether the attractors of   ∘   and   ∘   we are considering are linked or not.
In any case, it should be noticed that even when   ∘   and   ∘   have just one attractor, the map  , may have several of them depending on their distribution in the  plane (see, e.g., [18,19] and the examples of attractors in Figure 12).
Obviously, if   ∘   and   ∘   have 2 attractors, the number of different attractors of  , increases.So, still we have a practical work to do to decide whether   ∘   and   ∘   have one or two attractors.We fix the map   ∘   and realize that if it has three or four monotone pieces, then the image of its turning points is in {0, 1}.We take and converge it to where  and ] are mutually independent ergodic measures of   ∘   (see, e.g., [20]).Since different ergodic measures must be supported in different attractors, when the expression (27) tends to zero when  tends to infinity we have an evidence that just one attractor exists.Hence, we make simulations to compute (27) showing the results in Figure 8.Note that we have to be cautious when 0 or 1 is eventually periodic, because in such case its trajectory ranges all the attractors if the periodic orbit is an attractor, which need not happen (e.g., we can address that the well-known example of the map () = 4(1 − ) has a dense orbit, and hence, its attractor is the whole interval, and the image of its turning point is 1, which is eventually the fixed point 0).Fortunately, the probability of finding such pathological examples is 0, but this is the case for  = 2 and  > 1.
Taking into account the results obtained for generating Figure 8, we see that for  = 1.3 there are values of  around Figure 17: Average profit for firm 2 with different initial conditions.Note that in projections, the darker the color is, the lower the average profit is.
1.5 for which it seems that two different attractors of  , may coexist.Figure 9 shows a refinement of Figure 8 for  = 1.3 and 1.35.The bifurcation diagrams (see Figure 10) also suggest the existence of two different attractors for  = 1.3 but not for 1.35.
As an example, we show different types of attractors and limit sets (but nonmetric attractors) obtained for the parameter values  = 1.3 and  = 1.56 (see Figure 11 and the explanations therein).
When  = 1.35 the map   ∘   seems to have only one attractor, and, for  = 1.2, we see in Figure 12 that several of them may exist (see [18] for an analytical explanation of this fact).Figure 13 shows different types of attractors that can appear when we fix  = 1.86.
Once we have described the attractors of  , , we go back to the idea of explaining when topological chaos cannot be physically observed.To that end, it is worth to notice that for a  1 interval map , a periodic point  ∈  of period  is an attractor when |∏  =1   (  ())| ≤ 1, and therefore, its Lyapunov exponent (see, e.g., [14]) is  there is nothing to do.So, we assume that it is positive and let  1 , . . .,   be periodic intervals such that ∪  =1   is an attractor of type (A3).Any ergodic measure  supported on ∪  =1   must satisfy (  ) = 1/,  = 1, . . ., , and then we can consider   , , the invariant subinterval   for some 1 ≤  ≤ , and the ergodic measure (for   , ) ] given by the formula ]() = ( ∩   ) ⋅ .Since   , has negative Schwarzian derivative, by [23], we conclude that ], and therefore, , can be chosen to be absolutely continuous with respect to the one-dimensional Lebesgue measure.
For instance, we consider the values  = 1.3 and  = 1.56.The possible attractors were studied in detail in Figure 11, and we obtained the following average profits for the initial values as we showed in Table 1.
We compute the average profit for each firm.This result could be different if we change initial conditions when the number of attractors is different from one, as we studied in the previous section.Thus, for each firm we are going to present Table 1: We compute average profits for the initial conditions considered in Figure 11 and following the iteration processes we make there we check that both firms should produce starting from the initial conditions on the basis of the attractor ().Note that the profits in the fixed point (Cournot equilibrium) are 0.0140236 for firm 1 and 0.148848 for firm 2, and therefore, the only firm interested in stabilizing the Cournot point will be firm 2. When there are more than a fixed point we determine in which fixed point the profit is maximum for each firm.Finally we compare the average profit with the profit in the fixed point (in which it is maximum) and compute the residual profit.

Initial conditions
Figure 15 shows the average profit of both firms.For that, we use orbits of length  = 10000 and, as initial conditions, (, ), where  is the minimum of   ∘   when it exists and 0.4, otherwise, and  is the maximum of   ∘   .We will denote this election of initial conditions as min-max conditions.When  is greater than one there are more turning points which eventually may produce other average profits as it is shown in Figure 16.Here, we show the profit of firm 1, taking as initial conditions (, ), where  is the maximum of   ∘  and  is the maximum of   ∘  , which will be denoted as max-max conditions, and compare the situation with the initial conditions min-max previously computed.Figure 17 shows the same scenario for the second firm.
We are interested in analyzing whether the destabilization of Cournot points, even in the chaotic regime, is a good business for both firms.To this end, we compare the average profits of generic orbits with the expected profit at Cournot points.Firstly, we have to compute the number of fixed points of the composition   ∘  , which can be seen in Figure 18.The parameter space can be divided into two regions,  1 and  2 .In  1 there is one fixed point while in  2 the number of fixed points is three, which will be ordered  1 (, ) ≤  2 (, ) ≤  3 (, ).As an exception, for (, ) = (2, 2), the number of fixed points are four.In [1, Theorems 1 and 2] the stability and bifurcation diagrams of stationary points is studied in detail.
In region  2 we compute the profit of the first firm for each fixed point, and we observe that it is maximum for the fixed point  3 (, ) and minimum for  1 (, ), obtaining the converse situation for the second firm.Figure 19 (resp., Figure 20) shows the profit of the first firm (resp., second firms) against the profit obtained by the second firm (resp., second firm) when the fixed point that maximizes the profit of firm 1 (resp., firm 2) is chosen.
The case  = 1.3 was studied in Figure 11 and Table 1.We compute the profit of both firms in this case to compare the results see Figures 21 and 22.

Conclusions
A duopoly model depending on two parameters is studied in detail.By computing its topological entropy we characterize the parameter values which admit the existence of chaotic behavior of trajectories.We also analyze when the above mentioned chaotic trajectories either can be observed or remain on a set of zero Lebesgue measures and therefore are not physically observed in computer simulations.For that, we study the metric attractors of the model concluding that several of them can coexist.The existence of absolutely continuous ergodic measures is proved, and we showed that, even when chaotic maps are considered, the average of economic functions, like profit and production and along the orbits can attain a finite number of values.The existence of such values allows us to make decisions on whether the equilibrium points deserve to be stabilized by choosing strategies that are distinct from naive expectations.

Figure 1 :
Figure 1: Parametric regions for the composition maps  , (a) and  , (b) according to the number of pieces of monotonicity.

2. 3 .Figure 10 :
Figure 10: For  = 1.3,(a) and (b) show the bifurcation diagram for initial conditions 0 and 1, respectively.The bifurcation diagrams are constructed with generating orbits of length 100200 and representing just the last 200 points.(c) and (d) show the same when  = 1.35.

Figure 11 :
Figure 11: For  = 1.3 and  = 1.56, we make 200000 iterations and draw the last 100000 in the -axis.Except for (b) and (e), the initial conditions are taken to be the extremum of  , and  , .(a) and (e) are not attractors because the initial conditions are synchronized; in (a)   ( 0 ) =  0 , and in (e) (  ∘   ∘   ( 0 )) =  0 .These are -limit sets from initial conditions in a set of zero two-dimensional Lebesgue measures.If we change the initial conditions slightly, we obtain the attractors (b) and (f).Finally, (c) and (d) are two different attractors with the same projection in the and -axis, while for (b) and (f) the projections are different.Attractors (b) and (f) are possible because  , and  , have two different attractors consisting of two-periodic transitive intervals (of type (A3)).

Figure 14 :
Figure 14: We take orbits of length  = 10000 of   ∘   with initial conditions on the image of the turning points and estimate the Lyapunov exponent for (, ) ∈ [1, 2]×[0, 2] with step size of 0.01.In red we draw those values of the parameter for which the maximum of Lyapunov exponents for the turning points is smaller than or equal to 0.

Figure 15 :
Figure 15: Average profits of firm 1 (a) and firm 2 (b) for  = 10000 and step size for parameters  and  equal to 0.01.The initial conditions are  0 the maximum of   ∘   and  0 the maximum of   ∘   (type max-max).

Figure 16 :
Figure16: Average profit for firm 1 with different initial conditions.Note that in projections, the darker the color is, the lower the average profit is.
Average profit of firm 2 with min-max initial conditions and  ≥ 1 Average profit of firm 2 with max-max initial conditions and  ≥ 1 Projection of the average profit of firm 2 with min-max initial conditions Projection of the average profit of firm 2 with max-max initial conditions

Figure 19 :
Figure19: Average profits of firms 1 (a) and 2 (b) when we choose the fixed point which gives us the maximum profit of the first firm 1 among all the fixed points.Note that in projections, the darker the color is, the lower the profit is.

Figure 20 :
Figure20: Average profits of firms 1 (a) and 2 (b) when we choose the fixed point which gives us the maximum profit of the first firm 2 among all the fixed points.Note that in projections, the darker the color is, the lower the profit is.
3] by Adler, Konheim, and McAndrew.When  is Since turning points characterize the attractors of   ∘   and   ∘   , we conclude that attractors of type (A3) may exist if and only if (  ∘   )() > 0 for some turning point  of   ∘  .The following graphic shows the maximum Lyapunov