Market Share Delegation in a Bertrand Duopoly : Synchronisation and Multistability

This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their substitutability. This means that when products are complements, players may coordinate their behaviour even if initial conditions are different. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very different dynamic patterns.


Introduction
Strategic delegation is a relevant topic in both oligopoly theory and industrial organisation, and several papers have contributed to clarify questions related to the differences between the behaviour of profit-maximising firms and managerial firms (e.g., [1][2][3][4]).In the former kind of firms, ownership and management coincide, and consequently the main aim they pursue is profit maximisation.In the latter, ownership and management are separate and managers may be driven by incentive schemes that only partially take into account profit and the other objectives of the firms, such as output, revenues, relative performance evaluation, and market share [3,[5][6][7][8][9][10].In addition to the above-mentioned theoretical papers, there also exist some empirical works that stress the importance of market share delegation contracts in actual economies [11,12].
The present paper studies a nonlinear duopoly game with price competition and market share delegation and extends the study carried out by Fanti et al. [13] to the case of complementary or independent products.To this end, by following an established literature led by Bischi et al. [14], we assume that players have limited information and analyse how a managerial incentive scheme based on market share affects the local and global dynamics of a two-dimensional discrete time system.The paper stresses the differences with the analysis carried out by Fanti et al. [13] on the substitutability between products in the case with managerial firms and market share contracts and compares the results achieved.
The rest of the paper is organised as follows.Section 2 describes the model.Section 3 shows some preliminary global properties of the two-dimensional dynamic system (feasible set).Section 4 studies the fixed points of the system, the invariant sets, and local stability.Section 5 is concerned with multistability and shows that synchronisation may arise when managers receive the same bonus.It also stresses the differences with Fanti et al. [13] and takes into account the asymmetric case in which bonuses are not equally weighted in the managers' objective function.Section 6 outlines the conclusions.

The Model
Consider a duopoly game with price competition, horizontal differentiation, and market share delegation contracts (see [13] for details).Market demands of goods 1 and 2 are, respectively, given by where  ∈ (−1, 0] is the degree of differentiation of (complementary) products, while   ≥ 0 and   ≥ 0 are quantity and price per unit of good of firm  ( = 1, 2).
Both the firms have the same marginal cost 0 ≤  < 1 and hire a manager, who receives a bonus based on market share   /(  +   ) (,  = 1, 2,  ̸ = ), where   +   is total supply.The objective function of manager  is where Π  = (  − )  are profits and   > 0 is the (constant) delegation variable of player .Hence, by using ( 1), (2) becomes ,  = 1, 2,  ̸ = , from which we get the following marginal bonus: We now assume a discrete time ( ∈ Z + ) dynamic setting, where each player  has limited information, as in Bischi et al. [14], and uses the following behavioural rule to set the price for the subsequent period: where  > 0. We want to describe the qualitative and quantitative long-term price dynamics when products are complementary or independent, that is,  ∈ (−1, 0], and underline the similarities and differences with the case of substitutability investigated in Fanti et al. [13]. Assume   =  1,+1 ,  =  1, ,   =  2,+1 , and  =  2, .By using (4) and ( 5), the two-dimensional discrete time dynamic system is as follows:
The feasible set of system  is depicted in white in Figures 1(a) and 1(b) for two different parameter constellations, while the unfeasible points belonging to  are depicted in grey.The following evidence can be immediately observed: similarly to the substitutability case, (i) set  is nonempty such that  ⊂  and (ii) set  may have a simple structure (as in Figure 1(a)) or a complex structure (as in Figure 1(b)).The first evidence can be easily demonstrated by considering that the origin is a feasible point and that there exists a  > 0 small enough such that (1 − , 1 − ) is not a feasible point.
With regard to the study of the structure of the feasible set, a numerical procedure based on the study of the properties of the critical curves can be used (see, e.g., [15][16][17]).By taking into account the results proved in Fanti et al. [13], it is easy to verify that  is of  4 −  2 −  0 type since  can be subdivided into regions whose points have 4, 2, or 0 preimages, and the boundaries of such regions are characterised by the existence of at least two coincident preimages.
We denote the critical curve of rank-1 by LC (it represents the locus of points that have two or more coincident preimages) and the curve of merging preimages by LC −1 ; that is, the set of points (, ) ∈  such that |(, )| = 0, where is the Jacobian matrix of system  (see Figures 1(c) and 1(d)).
The study of the structure of set  is of interest from both economic and mathematical perspectives, since the long-term evolution of the economic system becomes pathdependent, and a thorough knowledge of the properties of  becomes crucial in order to predict the system's feasibility.
We now fix all parameter values but .Then, as is shown in Figures 2(a) and 2(b), if  = −0.15,set  has a simple structure (connected set), while when  = −0.2,set  consists of infinitely many nonconnected sets.This is due to the fact that the LC  curve moves upwards as parameter  decreases and consequently a threshold value  ≃ −0.1551 does exist such that a contact between a critical curve and the boundary of the feasible set occurs.This contact bifurcation causes the change of  from a connected set to a nonconnected set.In fact, a portion of the unfeasible set enters in a region characterised by a high number of preimages so that new components of the unfeasible set suddenly appear after the contact (in Figure 2(c), the feasible set is depicted immediately after the contact bifurcation creating grey holes inside the white region).The complexity of the structure of the feasible set increases if  decreases and the grey area increases too (i.e., the set of initial conditions generating unfeasible trajectories); however, it can be also observed that if  decreases further, a final bifurcation occurs.In fact, as  crosses a value  ≃ −0.218, almost all trajectories become unfeasible (see Figure 2(d)).
Notice that in the simulations presented in Figure 2 we have assumed  1 ̸ =  2 .However, a similar behaviour holds also when  1 =  2 : in this case, the contact bifurcation between the critical set and the boundary of the feasible set occurs at two points that are symmetric with respect to the main diagonal, so that the resulting feasible set is symmetric too (see Figures 1(a) and 1(b)).The previous arguments show that the bifurcations concerning the structure of the feasible set is strictly related to the value of the two key parameters  and   , which represent the degree of horizontal product differentiation and the level of market share bonuses.The following results can be proved.

Proposition 1.
Let  be the dynamic system given by (6).
According to Proposition 1, if   is high enough or products tend to be complements ( is low enough), the feasible set is consists of only the origin.This result confirms the one obtained in Fanti et al. [13] for the substitutability case; that is, economic meaningful dynamics are produced only when the degree of complementarity or substitutability between products is not too high.More precisely, in the case under investigation, economically meaningful longterm dynamics can be produced only for  ∈  − (0) and   ∈  + (0) ( = 1, 2), thus confirming the numerical experiments previously presented.
By taking into account the above-mentioned arguments, in what follows we will focus on the study of the dynamics produced by  by assuming that   is sufficiently small and  is sufficiently high.

Fixed Points, Invariant Sets, and Local Stability
Let  be the dynamic system given by ( 6) and consider a feasible initial condition.We now recall the results proved in Fanti et al. [13] concerning fixed points and other invariant sets of  for  ∈ (0, 1).It can be verified that they still hold also when  ∈ (−1, 0].We here present a sketch of the proof and we refer to Fanti et al. [13] for a further discussion. Remark 2. Let  be given by ( 6).Then, one has the following.))] and they can be complex; furthermore, Δ can be an attracting set.
(iii) The origin  0 = (0, 0) is a fixed point for all parameter values; it can be a stable node, an unstable node, or a saddle point.Up to two more fixed points on   and   may be owned.They are given by  1 = (0,  0 ) and  2 = ( 0 , 0) and can be unstable nodes or saddle points.(iv can be a stable node, an unstable node, or a saddle point. Proof.Consider the system  given by ( 6).
(i) (, 0) = (  , 0) and (0, ) = (0,   ); that is,   and   are invariant, and consequently the dynamics of  on such lines are governed by the two one-dimensional maps   = (, 0) and   = (0, ).As both   and   are unimodal for suitable parameter values (as proved in [13]), complex dynamics can be produced.  and   are repellor as the eigenvalue of (, 0) and (0, ) associated with the direction orthogonal to each semiaxis is greater than one for all parameter values.
(ii) Assume  1 =  2 = ; then, (, ) = (  ,   ), and hence Δ is invariant and the dynamics of system  on Δ are governed by () = (, ).Given the properties of () studied in Fanti et al. [13], nonmonotonicity can occur for suitable parameter values and complex dynamics may emerge.Finally, Δ can be an attracting set as the eigenvalue of (, ) associated with the eigenvector orthogonal to the diagonal may be less than one in modulus (see again Fanti et al. [13]).(iii) Since (0, 0) = (0, 0),  0 is a fixed point; furthermore, while considering (0, 0), it can be observed that  0 can be a stable node, an unstable node, or a saddle point.By solving   =  and   =  and following Fanti et al. [13], it can be easily verified that up to two positive solutions exist, namely,  1 and  2 ; such fixed points cannot be stable nodes as   and   are repellor.(iv) Assume  1 =  2 = ; then, by solving equation () =  it can be verified that it admits a unique positive feasible solution  *  iff  <  (see Fanti et al. [13] for more details); as each eigenvalue of ( *  ,  *  ) can be greater or less than one in modulus, then  *  can be a stable node, an unstable node, or a saddle point.
The question of the existence of an interior fixed point in the general case  1 ̸ =  2 (i.e., with different market share bonuses) cannot be addressed analytically.However, it will be discussed later in the paper by using numerical techniques.By taking into account parts (i) and (iii) in Remark 2, in what follows we will focus on the study of the dynamics produced by  for a feasible initial condition belonging to the interior of , thus focusing on economically meaningful initial states.

Synchronisation and Multistability
In order to study the evolution of system  when products are independent or complementary and compare this case with that of substitutable goods, we concentrate on the particular case of identical market share bonuses; that is,  1 =  2 = .As a consequence, system  in (6) takes the symmetric form   given by Since map   is symmetric, that is, it remains the same when the players are exchanged, then either an invariant set of the map is symmetric with respect to Δ or its symmetric set is invariant.By considering part (iv) in Remark 2, the local stability analysis of the unique interior fixed point  *  can be carried out by considering the Jacobian matrix associated with system   given by ) . ( Let Then, the Jacobian matrix evaluated at a point on the diagonal Δ is of the kind so that the eigenvalues of   (, ) are both real and given by while the corresponding eigenvectors are, respectively, given by V ‖ = (1, 1) and V ⊥ = (1, −1).
The eigenvalues evaluated at the fixed point  *  are, respectively, Thus,  *  can be attracting for suitable values of parameters such that both  ‖ ( *  ) and  ⊥ ( *  ) belong to the set (−1, 1).Different from the case in which products are substitutes, the following Proposition can easily be verified.
The following condition for the local stability of  *  (which holds if products are substitutes) applies also to the case  ∈ (−1, 0] and it can be recalled below (see Fanti et al. [13] for the proof).
Proposition 4. Let system   be given by (9).Then a  > 0 does exist such that  *  is locally asymptotically stable ∀ ∈ (−, ), given the other parameter values.
An attractor  located on the invariant set Δ exists for   if products are not too complementary ( is not too low) and the market share bonus  is not too large (see Figures 3(a) and 3(c)).This fact can also be confirmed by considering the following properties of map (): (0) = 0, lim  → 1 − () = −∞,   (0) > 1 ∀ ∈ (0, ),   () < 0 ∀ ∈ [0, 1).As a consequence, there exists a unique  − ∈ (0, 1) such that ( − ) = 0 and a unique maximum point   ∈ (0,  − ) such that (  ) is the maximum value of  in [0, 1).This implies that  is unimodal in [0,  − ].By considering that (  ) increases when  decreases and (  ) → +∞ if  → −1, then a  does exist such that (  ) >  − if −1 <  <  (see Figure 3(b)).On the other hand, by considering the role of parameter , a cascade of period doubling bifurcations is observed when  decreases (see Figure 3(c)).By taking into account the previous results and looking at the one-dimensional bifurcation diagrams in Figures 3(a) and 3(c), it can be observed that synchronised trajectories converge to the unique interior fixed point if  is close to  (the manager bonus is close to its upper limit).In addition, similar to what occurs with the logistic map, cycles can emerge due to period doubling bifurcation of  when  decreases (i.e., the degree of complementarity between products increases).This evidence is also confirmed when products are substitutes, thus proving that complexity in synchronised trajectories arises when moving away from the hypothesis of independent products.
Let us consider now a duopoly with identical players that start from different feasible initial conditions and let  ⊆ Δ be an attracting set of .If  =  *  for  = 0, then by considering (13) and Proposition 3 the following proposition holds.According to Proposition 6, if synchronised trajectories converge to  *  for  = 0, then trajectories starting from feasible initial conditions close to it, with (0) ̸ = (0), are synchronised in the long term as long as  ∈  − (0).If  decreases further, the fixed point loses stability firstly along the diagonal.This contrasts with the result obtained in Fanti et al. [13] where products are substitutes.In fact, if  passes from zero to positive values, the fixed point first loses transverse stability and consequently the trajectories are not synchronised.Therefore, synchronisation in this model is strictly related to the assumption of complementarity between products.
If  consists of a -cycle, then, similarly to what happens for the fixed point, several numerical computations show that if  is a -cycle for  = 0, then the -cycle loses stability firstly along the diagonal when  decreases, so that synchronisation may occur.Different from the case of substitutability, this evidence confirms that when products are complements, players may coordinate their behaviour towards a situation in which prices are equal (and the market is equally shared).In our analysis it is also stressed that the emergence of synchronisation with negative values of  also confirms the result obtained in Fanti et al. [18] with profitmaximising firms.
Consider now a more complex situation; that is,  is a chaotic attractor on Δ.In order to study its transverse stability, it is possible to use the procedure proposed in Bischi et al. [14], Bischi and Gardini [19], and Bignami and Agliari [20].Recall that the transverse Lyapunov exponent is defined as follows: where  0 ∈  and   is a generic trajectory generated by .
If  0 belongs to a generic aperiodic trajectory embedded within the chaotic set , then Λ ⊥ is the natural transverse Lyapunov exponent Λ  ⊥ , where natural indicates that the exponent is computed for a typical trajectory taken in the chaotic attractor .Since infinitely many cycles (all unstable along the diagonal) are embedded inside the chaotic attractor , a spectrum of transverse Lyapunov exponents can be defined and the natural transverse Lyapunov exponent represents a sort of weighted balance between the transversely repelling and transversely attracting cycles.If all cycles embedded in  are transversely stable, that is Λ max ⊥ < 0, then  is asymptotically stable in the Lyapunov sense for the twodimensional map   .Nevertheless, it may occur that some cycles embedded in the chaotic set  become transversely unstable; that is, Λ max ⊥ > 0, while Λ  ⊥ < 0. In such a case,  is not stable in the Lyapunov sense but it is a stable attractor in the Milnor sense.If a Milnor attractor of   exists, then some transversely repelling trajectories can be embedded in a chaotic set which is attracting only on average.In addition, such transversely repelling trajectories can be reinjected toward Δ so that their behaviour is characterised by some bursts far from the diagonal, before synchronization or convergence towards a different attractor.This situation is called on-off intermittency.
In order to investigate the existence of a Milnor attractor , we numerically estimate the natural transverse Lyapunov exponent Λ  ⊥ , represented with respect to  in Figure 4(a) for a fixed negative value of .It is possible to observe that it can take negative values.As an example, we consider  = 2.07 at which Λ  ⊥ < 0 while Λ max ⊥ > 0 and the one-dimensional map  exhibits a 4-piece chaotic attractor. is an attractor of system   belonging to the diagonal (see Figure 4(b)), but a trajectory starting from an initial condition that does not belong to the diagonal has a long transient before converging to  (see Figure 4(c)).In fact, by considering the difference () − () for any  we can observe that the transient part of the trajectory is characterised by several bursts away from Δ.The typical on-off intermittency phenomenon occurs.The whole trajectory starting from (0) = 0.1 and (0) = 0.2 is shown in Figure 4(d).
The study of the geometrical properties of the critical lines may be used to estimate the maximum amplitude of the bursts by obtaining the boundary of a compact trapping region of the phase plane in which the on-off intermittency phenomena are confined.Following Mira et al. [21], we obtain the boundary of the absorbing area in Figure 4(e) for the case presented in Figure 4(d).Observe that such a region contains the whole trajectory presented in Figure 4(d).However, not all trajectories are synchronised as   also admits a coexisting attractor, that is, a 2-period cycle whose basin is depicted in orange in Figure 4(f).
If   admits an attractor  ⊂ Δ and there exist feasible trajectories starting from interior points that are not synchronised, then the question of multistability has to be considered.In fact, several attractors may coexist (each of which with its own basin of attraction) so that the selected long-term state becomes path dependent, as in the situation shown in Figure 4(f).In this case, the structure of the basins of different attractors becomes crucial to predict the longterm outcome of the economic system.In Figure 5(a), the unique Nash equilibrium is locally stable, as the market share bonus is close to its upper limit.It is also globally stable, in the sense that it attracts all feasible trajectories taken into the interior of the feasible set  (that represents economic meaningful initial conditions).
If we compute /, then it is possible to observe that  increases as  decreases, and  → +∞ as  → −1 + .As a consequence, in order to obtain the convergence to the unique Nash equilibrium,  must be set at higher levels as the degree of horizontal product differentiation  decreases (i.e., products tend to be more complementary).Furthermore, if products are complements, then  ‖ crosses −1 as  decreases and a flip bifurcation occurs along the invariant set Δ; that is, a 2-period cycle appears close to the fixed point and it is stable along the diagonal.By taking into account the result proved in Proposition 3, when this bifurcation occurs,  ⊥ is still smaller than 1 in modulus and consequently (immediately after the first flip bifurcation) the 2-period cycle attracts all interior feasible trajectories.However, if  is still decreased, a period doubling bifurcation cascade occurs along the diagonal, so that more complex bounded attractors (such as periodic cycles) may exist on Δ around the unstable Nash equilibrium.As a consequence, the long-term synchronised dynamics may be characterised by bounded periodic (or even aperiodic) oscillations around the Nash equilibrium.If the attractor  is transversely unstable, the situation may become very complicated, as it is shown in Figure 5(b) where a 2-period cycle attracts all synchronised trajectories, while two attractors coexist out of the diagonal: a 2-period cycle  whose basin is depicted in orange and a 4-period cycle whose basin is depicted in yellow; note that the periodic points are in symmetric position with respect to the diagonal.As the parameter  decreases, a further flip bifurcation occurs and a 4-period cycle is created on the diagonal (see Figure 5(c)); furthermore, the 4-period cycle existing out of the diagonal becomes a stable focus and then undergoes a Neimark-Sacker bifurcation at which it becomes a 4-cyclic attractor formed by a 4-piece quasi-periodic attractor (green basin) coexisting with a 2-period cycle (orange basin), while synchronisation is avoided.This is how the situation presented in Figure 4(f) is approached: a final bifurcation that causes the transition to more complex basin boundaries occurs.Consequently to the final state sensitivity it is impossible to predict the long-term outcomes of the economy.
Finally, we consider the case in which products are independent from each other and each manager behaves as a monopolist ( = 0).When a flip bifurcation along the diagonal creates a -period cycle, it can be observed that a -period cycle is simultaneously created out of the diagonal as the eigenvalues of cycles embedded into the diagonal are identical.As a consequence, any period doubling bifurcation along Δ is associated with a period doubling bifurcation orthogonal to Δ.A similar phenomenon of multistability is presented in Bischi and Kopel [22].This case is shown in Figure 5(d): the green points represent initial conditions converging to a 4-period cycle on the diagonal while the yellow points represent initial conditions converging to a 4-period cycle out of the diagonal.This scenario occurs when the 2-period cycle along the diagonal undergoes the second period doubling bifurcation: two stable 4-period cycles are created, one along the diagonal (red points) and one with periodic points symmetric to it (black points); these two stable cycles coexist with the 2-period cycle previously created out of the diagonal (white points).Observe that an economy starting far away from the diagonal may become synchronised, as the basin of attraction  on the diagonal is comprised of several nonconnected sets.This result is relevant from an economic point of view.In fact, it implies coordination even though the manager hired in each firm behaves as a monopolist in his own market.

The Asymmetric Case.
We now consider the case in which managers' bonuses are evaluated differently; that is,  1 ̸ =  2 .Obviously, in this case Δ is no longer invariant (i.e., if the firms start from the same initial feasible condition ((0), (0)) ∈ , they will behave differently in the long term), so that synchronisation cannot occur.However, similar to the case in which products are substitutes, multistability still emerges.Assume  1 =  and  2 =  + Δ, where Δ ∈ (−, +∞).
With regard to the existence of the Nash equilibrium, we recall that Proposition 1 states a necessary condition; that is, parameter  should not be too close to its extreme value −1 and, in addition,  and Δ should not be too high.A Nash equilibrium, if it exists, is given by a point ( * ,  * ) ∈  such that ( * ,  * ) = ( * ,  * ) = 0.As a consequence, an interior fixed point can be obtained by considering the intersection points of the two curves (, ) = 0 and (, ) = 0 in the phase plane.Of course, if these curves intersect in a point  * = ( * ,  * ) ∈ , then it is a Nash equilibrium for .
By considering the analytical properties of  and , numerical simulations allow us to conclude that the main results of Fanti et al. [13] are confirmed also when products are complements or independent.These results are collected in the following list.
(i) If the Nash equilibrium exists, then it is unique such that the equilibrium price is higher for the variety associated with a lower market share bonus (see Figure 6(a)).
(ii) If the Nash equilibrium is locally stable in the symmetric case, then it is also locally stable in the asymmetric case if and only if the perturbation on  is small enough (i.e., Δ is close to zero).
(iii) In the case of heterogeneity, the Nash equilibrium loses stability via a flip bifurcation at which it becomes a saddle point and a stable 2-period cycle appears close to  * (see Figure 6(b)).
(iv) Synchronised trajectories do not emerge and synchronisation cannot occur, while multistability still emerges (compare Figures 6(c) to 4(f)).
To better describe point (iv) above, we recall that a situation in which synchronisation may occur is depicted in Figure 4(f): the symmetric system   admits a complex attractor on the diagonal that coexists with a 2-period cycle out of the diagonal.If we consider a slight difference between weights attached to market share bonuses, that is, Δ is small enough, we obtain the situation depicted in Figure 6(c): the attractor on the diagonal disappears while a complex attractor coexists with the 2-period cycle previously found.
Similarly, observe that, with independent products and homogeneous managers, three coexisting attractors are owned (see Figure 5(d)).This situation drastically changes if Δ = 0.01.In fact, as shown in Figure 6(d), a small perturbation on  causes the disappearance of the attracting 4-period cycle symmetric to the diagonal, while a 4-period cycle close to the diagonal persists together with the 2-period cycle existing out of the diagonal.Obviously, due to the heterogeneity between the weights   , the shape of the boundaries of the coexisting attractors is no longer symmetric with respect to the diagonal.
Although the managers' behaviours are no longer coordinated, it is interesting to stress that, different from the case of substitutability between products, the structure of the basin of attraction seems to become simpler than under homogeneous delegation contracts.

Conclusions
This paper has studied the mathematical properties of a nonlinear duopoly game with price competition and market share delegation contracts.The main aim was to extend the analysis carried out by Fanti et al. [13] to the case in which products are complementary or independent.The most important result is that the interaction between the degree of complementarity and the delegation variable (which affects managerial bonuses) may produce synchronisation in the long term.This result does not emerge when products are substitutes.
From an economic point of view, synchronisation is relevant because it implies coordination between players.Then, in a model with managerial firms and market share delegation contracts, coordination can (resp., cannot) hold when products are complements (resp., substitutes).In addition, we have also shown that multiple attractors may exist so that initial conditions matter.

Figure 3 :
Figure 3: Parameter values:  = 0.5,  = 1.5.(a) One-dimensional bifurcation diagrams with respect to  for two fixed  values.(b) Map  is plotted for different  values and  = 1.(c) One-dimensional bifurcation diagrams with respect to  (0 <  < ) for two fixed  values.

Figure 4 :
Figure 4: Parameter values:  = 1.5,  = −0.2, and  = 0.5.(a) The natural transverse Lyapunov exponent with respect to parameter .(b) Four-piece chaotic attractor  of system   belonging to the diagonal for  = 2.07.(c) Bursts away from the diagonal before synchronization for  = 2.07, (0) = 0.1, and (0) = 0.2.(d) The whole trajectory starting from initial condition as in (c) and converging to the attractor in (b).(e) The minimal absorbing area in which on-off intermittency phenomenon occurs for the same parameters as in (c).(f) The attractor  coexists with an attracting 2-period cycle for  = 2.07.

Figure 5 :
Figure 5: Parameter values:  = 1.5 and  = 0.5.(a) The white region represents the basin of  *  for  = 5.1 and  = −0.2.(b) A 4-period cycle together with a 2-period cycle coexists with the attractor  belonging to the diagonal (a two-period cycle) for  = 2.4 and  = −0.2.(c) For  = 2.3, a 4-piece quasi-periodic attractor has been created,  = −0.2.(d) A particular scenario is presented for  = 0.2 in the case of independent products  = 0.