Hidden and self-exited attractors in a heterogeneous Cournot oligopoly model
HHidden and self-exited attractors in a heterogeneous Cournotoligopoly model
Marius-F. Danca a,b, ∗ , Marek Lampart c,d a Dept. of Mathematics and Computer Science, Avram Iancu University of Cluj-Napoca, Romania b Romanian Institute of Science and Technology, 400487 Cluj-Napoca, Romania c IT4Innovations, VSB - Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava, CzechRepublic d Department of Applied Mathematics, VSB - Technical University of Ostrava, 17. listopadu 2172/15, 70833 Ostrava, Czech Republic
Abstract
In this paper it is numerically proved that a heterogeneous Cournot oligopoly modelpresents hidden and self-excited attractors. The system has a single equilibrium and aline of equilibria. The bifurcation diagrams show that the system admits several attractorscoexistence windows, where the hidden attractors can be found. Intensive numerical testshave been done.
Keywords:
Hidden attractor; Self-excited attractor; Cournot oligopoly model
1. Introduction
Since 1838, when A. Cournot [7] proposed the first treatment of oligopoly (a duopolycase), the theory of a market form in which a market has a dominant influence on a smallnumber of sellers (oligopolists) was deeply researched. The first and crucial additions tothe theory were made by H. von Stackelberg [24] and later on significances to the theorywere done from a different point of view.As the Cournot-Nesh equilibria, of the corresponding game, reflects given oligopolybehaviour, its stability has to be investigated depending on the number of players and alsoon the way of the game modelling. For the second case, it was given by [23] (see also [19]page 237) that the oligopoly model constructed under constant marginal costs with a lineardemand function is neutrally stable for three competitors and unstable for more than threecompetitors (for more details see [22]). It is noted in [22] that linear demand functions arevery easy to use, but they do not avoid negative supplies and prices, so it is possible touse them only for the study of local behaviour. Hence, nonlinear demand functions such ∗ Corresponding author
Preprint submitted to Elsevier July 2, 2020 a r X i v : . [ n li n . C D ] J u l s piecewise linear functions or other more complex functions were applied. For duopolyby [20], and later by [21] for a triopoly using iso-elastic demand functions. These typesof demand function were later studied by [1] and [2] for a nonlinear (iso-elastic) demandfunction and constant marginal costs and it was concluded that this Cournot model for n competitors is neutrally stable if n = 4 and is unstable if the number of competitors isgreater than five (see also [22]). Finally, a complete characterization of the Cournot-Neshstability was done in [16] depending on the number of competitors.All the above-given approaches were done for a homogeneous approach. In heteroge-neous decision mechanism, introduced in [5], two different types of quantity setting playerscharacterized by different decision mechanisms that coexist and operate simultaneouslyare considered. In this case, competitors adaptively get used their choices towards thedirection increasing their profits. This model’s Cournot-Nesh equilibria stability was de-scribed showing is periodic and also chaotic regimes. Moreover in [6], an addition to theforegoing approach was done where the role of the intensity of scenario choice was takeninto consideration.On the other side, hidden attractors represent an important recently introduced notionin applications because they might allow unexpected and potentially disastrous systemsresponses to some perturbations in a structure like a bridge or aircraft wing. However,except some examples of theoretical models (see e.g. [11, 14, 13]), there are no importantinvestigations on hidden attractors in real and applied examples of chaotic maps.The generally accepted definition which gives an attractors classification is Definition 1. [15, 17] An attractor is called self-excited attractor if its basin of attrac-tion intersects with any open neighborhood of an equilibrium, otherwise it is called hiddenattractor.
Sudden appearance of some hidden chaotic attractor, could represents a major disad-vantage for the underlying system. Thus, the consequences could be dramatic such asin the case of pilot-induced oscillations that entailed the YF-22 crash in April 1992 andGripen crash in August 1993 [4]. It is understandable that identifying unwanted hiddenchaotic behavior is a desirable phenomena. There exists the risk of the sudden jump froma desirable attractor to possible undesired behavior of some hidden attractor. Recently, ithas been shown that multistability is connected with the occurrence of hidden attractors.If there are unstable fixed points, the basins of attraction of the hidden attractors do nottouch them, being located far away from such points. Note that if the system exhibitsa chaotic or regular behavior while systems equilibria are stable, then the chaotic or reg-ular underlying attractors are implicitly hidden. Therefore, the stability of equilibria isimportantFor a hidden attractor, its attraction basins are not connected with unstable equilibria.Hidden attractors can be found in e.g. systems with no-equilibria or with stable equilibria[4]. 2lso, as in the case of the studied discrete-time system in this paper, systems withinfinite number of equilibria (also called line of equilibria), can admit hidden attractors.Systems with a line of equilibria are very few (see e.g. [12, 18]). Hidden attractors intoan impulsive discrete dynamical system have been found in [9], where the case of a supplyand demand economical system is studied.The paper is organized as follows: Section 2 presents the considered oligopoly model,underlining equilibria stability, necessary in the study of hidden attractors, Section 3 dealswith hidden and self-exited attractors, while Conclusion ends the paper.
2. The heterogeneous Cournot oligopoly model
Consider the heterogeneous Cournot oligopoly model (HCOM) introduced in [6] definedfor identical quantity setting agents N = { , , . . . , N } that compete in the same marketfor an homogeneous good, whose demand is summarized by a linear inverse-demand func-tion, or price function P ( Q ) = max { a − bQ, } ( P treats price as a function of quantitydemanded). Denote by q ni the quantity of goods that is generic i -th agent, with i ∈ N , sellsin the market at time-period n . All the agents bear the same constant marginal productioncost c , so that the generic i -th agent earns the profit π i = P ( Q ) q i − cq i . The oligopoly in this case, is characterized by introducing heterogeneous decision mech-anisms, used to decide how much quantity of goods to produce, by considering a populationstructured into two groups of agents of different kinds. The first group whose represen-tative is denoted by q includes boundedly rational players that use gradient rule, that iscalled gradient player. The second group with marked representant q includes agents thatadopt an imitation-based decision mechanism called imitator players.The collective behavior of the whole heterogeneous population of N players is describedby the following 2-dimensional non-linear autonomous discrete dynamical system [6]:HCOM : q n +11 = q n + γq n ( a − b (( N (1 − ω ) + 1) q n + ωN q n ) − c ) ,q n +12 = π n π n + π n q n + π n π n + π n q n , (1)where π n = ( a − c − bN ((1 − ω ) q n + ωq n )) q n , (2) π n = ( a − c − bN ((1 − ω ) q n + ωq n )) q n , (3)The system admits a line of equilibria X ∗ (0 , q ) , q ≥ , Oq ), and also the single equilibrium X ∗ (cid:16) a − cb ( N + 1) , a − cb ( N + 1) (cid:17) . The stability of equilibria is stated by the following result
Theorem 2. [6]i) Equilibrium X ∗ is asymptotically stable if ω ∈ (cid:0) Ω , Ω (cid:1) , with Ω = 32 N + 1 N (cid:16) − γ ( a − c ) (cid:17) , Ω = 12 N + 1 N (cid:16) γ ( a − c ) + 1 (cid:17) . ii) Equilibria X ∗ (0 , q ) are stable for q > q ∗ = a − cbωN . Note that the stability, established by Theorem 2 is locally . Graphical interpretation
Consider in the parameter space ( γ, ω ), the latice domain D = [0 . , . × [0 . , . M (0 . , . , M (cid:48) (0 . , . , M (cid:48) (0 . , . , M (0 . , . a = 10, b = 1, c = 1, N = 5 and the parameters γ , ω as bifurcation parameters. For all these values, theequilibrium X ∗ = (1 . , . γ , Ω , with graph Γ and Ω , with graph Γ , are bifurcation curvesrepresenting the flip bifurcation and Neimark-Saker bifurcation, respectively, of X ∗ . Forall ( q , q ∗ ), X ∗ suffers a Neimark-Saker bifurcation [6].The stability of points X ∗ , reads as follows: the dark green area S ⊂ D limited by thecurves Γ , and lines γ = 0 .
35 and γ = 0 .
525 and defined by the corners M (0 . , . ,M (0 . , . , M (0 . , . , M (0 . , . γ, ω ) which generate the stability of the equilibrium X ∗ , while the light green areas (outsidethe curves Γ , ) represent the instability sets of the equilibrium X ∗ .The vertical dotted line through γ = 0 .
48 and the horizontal dotted line through ω = 0 . ω and γ , respectively. In [6] this relation seems to be wrong. . Attractors coexistence and hidden attractors As know, due to the uniqueness of the solutions of the Initial Value Problem (IVP) (1)(ensured by the explicit form of the system equations), for a fixed parameter value, to eachinitial condition corresponds uniquely an attractor. In this paper one consider numericalattractors , obtained by numerical integration of the IVP, after sufficiently large transientsremoved. For simplicity, hereafter, by attractor one understands the underlying numericalattractor. Therefore, in a bifurcation diagram ( BD ) generated with fixed initial condition,to every bifurcation parameter value corresponds a unique attractor represented in the BD as a vertical line, composed from a set of isolated points (periodic attractors), or a bandof infinity of points (quasiperiodic or chaotic attractors) .Because in this paper every BD is generated with two different initial conditions ( IC ),namely IC = (1 , .
5) and IC = (1 . , . BD denoted with fraktur letters with index 1 and 2:( A , A ), ( B , B ) and so on. Therefore, at the considered resolution of 800 points on thebifurcation parameter axis, correspond two sets of 800 attractors, denoted with calligraphicletters indexed 1 or 2 depending the initial conditions IC , , A ∈ A , B ∈ B , A ∈ A , B ∈ B and so on. All these attractors are function of the considered bifurcationparameter, A = A ( p ), B = B ( p ) and so on, the parameter p being either γ or ω , andare plotted red and blue corresponding to IC , or to IC , respectively.For simplicity, hereafter one drops the parameter p in attractors notation and all at-tractors in this paper are generated by starting from one of the initial conditions IC , or IC . Note that every attractor can also be generated from the indicated initial conditionsfrom underlying attraction basins, denoted q .For some parameter ranges, the two sets of attractors could be different (inside the co-existence windows), when the existence of hidden attractors is possible, or identic (outsidecoexistence windows).Within the studied coexistence windows, the system (1) presents three different kindof attractors: periodic attractors or limit cycles, quasiperiodic attractors and chaotic at-tractors.The tools utilized in this paper to identify attractors are: BDs, time series, planarphase representations, the maximal local finite-time Lyapunov exponent λ , the output K of the 0-1 test for chaos (see e.g. [10]) and Power Spectrum Density (PSD). Because thePSD is two-sided symmetric, only the left-side is considered. The numerical integration ofthe system (1) has been effectuated for n = 3000 iterations.Following Definition 1, the algorithm used to detect numerically hidden and self-excitedattractors of the considered system (1) is presented in the diagram in Fig 2. In systems To be focused, in this paper every chaotic orbit is as usual understood as an orbit approaching a chaoticattractor, even if the exiting (interior and exterior) crises might imply chaotic but non-attracting sets (seee.g. the non-attracting chaotic set after the saddle-node bifurcation of the logistic map for r ≈ .
5n spaces with higher dimension with unstable equilibria, the attraction basins are chosenusually as planar sections containing unstable equilibria. The case of three-dimensionalneighborhoods of the Fabrikant-Rabinovich system is treated in [8]. As the diagram shows,the main steps in finding hidden attractors bases on testing if the analyzed attractor hasinitial conditions within no matter how small neighborhoods of all unstable equilibria ( X ∗ , X ∗ ). If there exists a neighborhood of at least one of the unstable equilibria containinginitial conditions of the considered attractor, the attractor is self-excited. Otherwise, if theattraction basin does not intersect any of unstable equilibria, the attractor is hidden.Because the case of stability of both equilibria is trivial (for example the case of chaoticattractors which, in this case, are all hidden by Definition 1), one considers the complicatedcase of unstable equilibria, when each equilibrium must be analyzed.The main step in verifying if the attractors are hidden or self-excited, following thealgorithm in Fig. 2, is to check neighborhoods of equilibria X ∗ , . Precisely, one hasto verify the connection of the attraction basins of the considered attractor with bothequilibria. For this purpose one examine neighborhoods of both equilibria X ∗ and X ∗ ,considered separately for clarity (see e.g. Figs. 6). The figures in Figs. 6 represent laticesof points ( q , q ), considered as initial conditions for numerical integration of the IVP (1),containing equilibria X ∗ and X ∗ , respectively. For X ∗ , which is a line of equilibria, theregion containing the equilibrium, is a rectangular neighborhood with width 1 e − q ∗ . Thus, the neighborhood contains a part of the lineequilibria X ∗ including the critical point q ∗ (see Theorem 2 ii)). Conform to Theorem 2ii) the yellow points with q > q ∗ , represent initial conditions leading to the vertical axis,which is attractive, while for q < q ∗ , X ∗ is repulsive. For X ∗ , the examined region is asquared lattice with side 1 centered on X ∗ . On both neighborhoods, red plot representsthe initial conditions leading to attractors of the first set, corresponding to IC , while blueplot are the points leading to attractors belonging to the second coexisting set of attractorscorresponding to IC . Black points represent the divergence points, for which the systemis unbounded. γ -attractors Consider the BD of q and q vs γ ∈ [0 . , . ω = 0 .
4, denoted with BD γ (dotted line through ω = 0 . γ = 0 .
4, the diagram crosses thecurve Γ (point F in Fig. 1), marking the first flip bifurcation of X ∗ , which at γ = 0 . BD s with respect γ and ω axis, for simplicity, hereafter onlythe component q is considered.As specified bellow, attractors belonging to A are denoted with A , while the attractorsof A are denoted with A (red and blue plot, respectively, in Figs. 4 (a), (b)).Because the system dynamics related to hidden attractors regard mainly unstable equi-libria, one consider on the BD γ the values γ > .
4, where the equilibrium X ∗ is unstable6see Fig. 1) and where there are the coexistence window denoted A and the zoomed win-dow B , delimited by γ ∈ (0 . , . γ ∈ (0 . , . A starts with an exterior crisis at γ = 0 . A , and ends with aninterior crisis at γ = 0 . B starts with an exterior crisis at γ = 0 . A , , the bifurcations of A being “delayed” withrespect γ , compared to the bifurcations of the attractor A . At γ = 0 .
508 the window endswith an interior crisis of the attractor A .Consider first γ = γ = 0 .
506 (Fig. 4 (a)) to which correspond two coexisting attractors, A ∈ A and A ∈ A . The type of the attractor A is revealed by the three red bulletpoints in BD γ (Fig. 4 (a)), λ , which is negative (bullet on the light green curve in Fig.4 (c)) and K which is 0 (bullet on the light magenta curve in Fig. 4 (c)). Moreover, thetime series (Fig. 5 (a)) and the phase plot (Fig. 5 (c)) indicate that the attractor A isperiod-3 (the three numbered red bullets in Fig. 5 (a) and (c)). The attractor A presentsa so called chaotic band (light blue band in Fig. 4 (a)). λ is positive (bullet on dark greencurve in Fig. 4 (c)) and K ≈ q axis (Fig. 5 (c)) indicate the chaotic characteristic of A .For γ = γ = 0 . A and A , are chaotic, as shown bythe BD γ , λ , K (Figs. 4 (c)), the time series in Figs 5 (d), (e) and the phase plots in Fig. 5(f). While the chaotic attractor A contains three chaotic bands (purple lines in Fig. 4 (a)and Fig. 5 (a), (f)), the chaotic attractor A is composed by a single chaotic band (lightblue in Fig. 4 (a) and Fig. 5 (e) and Fig. 5 (f)). Note that the three chaotic bands bornfrom the previous stable three red points in Fig. 5 (a), (c) which lost the stability.At γ = γ = 0 . A , and A , are two periodic attractors. The stable cycle A is a period-6attractor (the apparent cycle point at about q = 2, marked with * in the time series (Fig.5 (g)), is actually a superposition of two points of the cycle: points 2 and 5 in Fig. 5 (i)).The other stable cycle, A , is a period-7 attractor. The periods are revealed by the redand blue bullets, respectively, in the BD γ , negative λ and zero K (Fig. 4 (d)).The last considered value, γ = γ = 0 . A (thesix red bullets in Fig. 5 (j)) and the six pieces chaotic attractor A , which presents sixlight blue chaotic bands (see Fig. 5 (k) and the phase plot in Fig. 5 (l)). The periodiccharacteristic of A is underlined by the negative λ (light green, Fig. 4 (d)) and the zerovalue of K (light magenta, Fig. 4 (d)). Again, at q = 2 there are two overplotted points(3 and 6 in Fig. 5 (j) and (l)).The analysis made in Figs. 6 for γ -attractors, show that for every considered cases of γ ,the attraction basins of attractors A for γ ∈ { γ , γ , γ , γ } (red plot) have no connectionwith X ∗ or X ∗ and, therefore, they are hidden, while attractors A , for γ ∈ { γ , γ , γ , γ } ,have initial conditions which intersect any neighborhood of X ∗ (blue plot), and, therefore,7re self-excited.Note that, for the considered γ values, attractors A and A IC and IC , from indicated initial conditions q too. Also, the critical value of q ∗ is q ∗ ≈ . γ = γ , the period-3 stable cycle A is hidden, while the chaotic attractor A isself-excited (Fig. 6 (a));2. For γ = γ , the chaotic attractor A is hidden, while the chaotic attractor A isself-excited (Fig. 6 (b));3. For γ = γ , the period-6 stable cycle A is hidden, while the stable period-7 stablecycle A is self-excited (Fig. 6 (c));4. For γ = γ , the period-6 stable cycle A is hidden, while the chaotic attractor A isself-excited (Fig. 6 (d)). ω -attractors Consider the BD versus ω , BD ω , for ω ∈ [0 . , . γ = 0 .
48 generated withthe same initial conditions, IC and IC (Fig. 7). Like the BD γ , the BD ω crosses thestability and instability domains S and I (Fig. 1). Compared to the BD γ which intersectsonly the flip bifurcation curve, Γ , BD ω intersects the NS bifurcation curve, Γ , at points Q (cid:48) and Q (cid:48)(cid:48) , as well, the ingredient necessary to quasiperiodic oscillations. At ω = 0 . X ∗ becomes stable (Fig. 1).Note that for ω ∈ (0 . , . ω which starts with the last reverse flipbifurcation (point Q (cid:48) ) and ends at the first NS bifurcation (Point Q (cid:48)(cid:48) ), the system dynamicsdo not depend on ω , fact which could represents a useful system characteristic. Denote thetwo coexisting sets of attractors with B and B with corresponding elements, attractors B and B , respectively.The considered coexistence windows starts with successive reversed flip bifurcations(period halving bifurcation) of the attractors B , while the attractors B remain chaoticfor a large parameter range ω ∈ (0 . , . B .The windows of interest generated by the parameter ω are denoted by C , his successivezoomed area D and E (Fig. 7 (a), (b), (c)).Within area C and his zoomed area D , one consider three representative cases: ω =0 . ω = 0 . ω = 0 . ω = ω , the values of λ , and K suggest chaotic dynamics for both underlyingattractors, B and B (Fig. 7 (c)). The time series in Fig. 8 (a) and the phase plot (Fig.8 (b)) shows that the attractor B produces three chaotic bands V (magenta plot). Theattractor B presents a single large chaotic band (see Fig. 8 (b) and phase plot in Fig. 8(c), light blue plot). 8or ω = ω , by a reverse flip bifurcation, the former attractor B , transforms into astable period-6 cycle (see Fig. 8 (d) and (f) where elements 1 and 4 have the same q value), while the attractor B remains chaotic but with a reduced size of the underlyingchaotic band (Fig. 8 (e), (f)).At the last considered value ω = ω , by reverse flip bifurcation both attractors, B andthe attractor B , transform in stable cycles of period-3 and a period-14 cycle, respectively(Figs. 8 (g), (h), (i)).To see which ω -attractors are hidden, one applies the algorithm presented in Fig. 2,for each considered case of ω .Again, for the considered ω values, attractors B and B can be generated, beside IC and IC , from indicated initial conditions q in the attraction basins too (Figs. 9). Thecritical value of q ∗ is q ∗ ≈ . B are hidden (the red plot corresponding to the ICs of the attractor B ,indicates that the attraction basins do not touch equilibria), while B are self-excited.1. For ω = ω , the chaotic attractor B is hidden, while the chaotic attractor B isself-excited;2. For ω = ω , the period-6 cycle B is hidden, while the chaotic attractor B is self-excited;3. For ω = ω , the period-3 cycle B is hidden, while the period-14 cycle B is self-excited; Quasiperiodic attractors
An interesting particular case of ω -attractors unlike the previous cases, is representedby the coexisting window E , defined by ω ∈ [0 . , . C and C with elements C and C , generated asfor all cases from q = (1 . , . ω = 0 .
745 (Fig. 10 (a)). In this case λ is negative for C andzero for C , and K is zero for both attractors (Fig. 10 (b)) fact which indicate that C isquasiperiodic, while C period-4 cycle. This conclusion is sustained by the four red bulletsin time series in Fig. 11 (a), and the grey quasiperiodic band in Fig. 11 (b) and also phaseplot in Fig. 11 (c). Note that the fact the points of the quasiperiodic curve in the phaseplot tend to fill the entire closed quasiperiodic orbit (invariant circle), indicates that theorbit neither closes nor repeats itself. The quasiperiodicity ie revealed also by the PSDwhich shows clearly that the periodic orbit ( C ) presents the first fundamental frequency f and the harmonic f situated at distance δ (Fig. 11 (d)). Regarding C , because ofthe NS bifurcation, which generally generates quasiperiodicity, a new set of subharmonicsborn, like f , close to first frequencies f and f , at smaller distance distances δ .The attractor C is hidden, while the quasiperiodic orbit C is self-excited (see theattraction basins in Fig. 12 (a)). Compared to previous cases, the attraction basins inthis case, have more complicated shape and remember the riddled attraction basins (see9 γ Hiddenattractors Self - excitedattractors F iguresω = 0 . γ = 0 . A period -3 A chaotic F igs. a ) − ( c ) γ = 0 . A chaotic A chaotic F igs. d ) − ( f ) γ = 0 . A period -6 A period -7 F igs. g ) − ( i ) γ = 0 . A period -6 A chaotic F igs. j ) − ( l ) ω = 0 . γ = 0 . B chaotic B chaotic F igs. a ) − ( c ) ω = 0 . B period -6 B chaotic F igs. d ) − ( f ) ω = 0 . B period -3 B period -14 F igs. g ) − ( i ) ω = 0 . C period -4 C quasiperiodic F igs.
11 ( a ) − ( c ) Table 1: Hidden and self-excited attractors of the system (1). . e.g. [3]), when any arbitrary neighborhoods of every points of the attraction basin seemsto contain points from some another basin (see e.g. circled regions of X ∗ , neighborhoods).As can be seen, there exist thin yellow strips of points ( q , q ) with q < q ∗ for which X ∗ is attractive, in contradiction with Theorem 2 ii). For example for the initial condition q = (0 . , . q = 0 . < .
42 = q ∗ , the orbit is attracted by thevertical axis of equilibria X ∗ (Fig. 12 (b)). The second component of the orbit tends to avalue situated beyond q ∗ , where X ∗ is stable (see q ≈ . X ∗ reveal the fact that q ∗ is not constant with respect q , but a functionof q too and, therefore, the graph of q ∗ (separatrix between the yellow and blue domain)is not a constant horizontal line. These apparent contradictions, could be related to thelocally character results of the stability given by Theorem 2.All results are presented in Table 1. Acknowledgements
This work was supported by The Ministry of Education, Youthand Sports from the National Programme of Sustainability (NPU II) project “IT4Innovationsexcellence in science – LQ1602”; by The Ministry of Education, Youth and Sports fromthe Large Infrastructures for Research, Experimental Development and Innovations project“IT4Innovations National Supercomputing Center – LM2015070”; by SGC grant No. SP2020/137“Dynamic system theory and its application in engineering”, VSB - Technical Universityof Ostrava, Czech Republic, Grant of SGS No. SP2020/114, VSB - Technical Universityof Ostrava, Czech Republic.
4. Conclusions
In this paper hidden and self-exited attractors of a discrete heterogeneous Cournotoligopoly model were numerically found. The system proved to have extremely rich dy-namics including attractors coexistence. All studied coexistence windows embed hidden10ttractors and self-excited attractors. To identify hidden attractors, one analyze the neigh-borhoods of unstable equilibria in order to see they have connections with the consideredattractors. 11 igure 1: The explored rectangular area M M (cid:48) M (cid:48) M on the parameters plane ( γ, ω ). Dark green area M , M , M , M represents the area where X ∗ is stable. Parameters points ( γ, ω ) within this area generatecounterclockwise orbits spiralling toward the stable equilibrium X ∗ . Light green represents the instabilitydomain of X ∗ . Curves Γ , represent the limits stability. Γ is the flip bifurcation curve and Γ the NSbifurcation curve. Horizontal dotted line through ω = 0 . γ , BD γ . The near red points represent the studied values of γ in this bifurcation diagram. Vertical dottedline through γ = 0 .
48 represents the bifurcation diagram versus ω , BD ω . Blue near points on this linerepresent the values of ω studied in this bifurcation diagram. Points F , Q (cid:48) , and Q (cid:48)(cid:48) , Q are points of flipand NS bifurcations, respectively. igure 2: Algorithm utilized in this paper to identify hidden attractors. igure 3: Bifurcation diagram of the system (1) versus γ , BD γ , for both components q , , for ω = 0 . igure 4: Zoomed areas of the bifurcation diagram BD γ . (a) Zoomed area for γ ∈ [0 . , . γ ∈ [0 . , . K and λ for the zoomed area A ; (d) K and λ for the zoomed area B . igure 5: (a) Time series of the periodic attractor A for γ = 0 .
506 with initial condition q = (1 . , . A for γ = 0 .
506 with initial condition q = (1 . , . A and A for γ = γ ; (d) Time series of the chaotic attractor A for γ = 0 . q = (1 . , . A for γ = 0 . q = (1 . , . A and A for γ = γ . igure 5: Continuation: (g) Time series of the periodic attractor A for γ = 0 . q = (1 . , . A for γ = 0 . A and A for γ = γ ; (j) Time series of the periodic attractor A for γ = 0 . q = (1 . , . A for γ = 0 . A and A for γ = γ . igure 6: Attraction basins of attractors A and A for the considered four γ values, considered aroundboth equilibria X ∗ , . Red area represents initial conditions of the attractor A , while blue parts of theattraction basin of A . Yellow points are attracted by X ∗ which, for q > q ∗ is attractive. Points fromblack area tend to infinity. (a) γ = γ ; (b) γ = γ ; (c) γ = γ ; (d) γ = γ . igure 7: (a) Bifurcation diagram of q versus ω , BD ω for γ = 0 .
48; (b) Zoomed area C of the BD ω for ω ∈ [0 . , . C ; (d) K and λ of the area C . igure 8: (a) Time series for the chaotic attractor B for ω = 0 . q = (1 . , . B for ω = 0 . B and B for ω = ω ; (d) Time series of the periodic attractor B for ω = 0 . q = (1 . , . B for ω = 0 . B and B for ω = ω ; (g) Time series of the periodic attractor B for ω = 0 . q = (1 . , . B for ω = ω with initial condition q = (1 . , . B and B for ω = ω . igure 9: Attraction basins of attractors B and B for the considered three ω values, considered aroundboth equilibria X ∗ , . Red area represents parts of the attraction basin of the attractor B , while blue partsof the attraction basin of B . Yellow points are attracted by X ∗ which, for q > q ∗ is attractive. Pointsfrom black area tend to infinity. (a) ω = ω ; (b) ω = ω ; (c) ω = ω . igure 10: (a) Zoomed area E of the BD ω for ω ∈ [0 . , . K and λ . igure 11: (a) Time series of the periodic attractor C for ω = 0 . C ; (c) Phase plot of the attractors C and C ; (d) PSD of the component q of the attractor C ;(e) PSD of the component q of the attractor C . igure 12: (a) Attraction basin of attractors C and C for ω = 0 . X ∗ , . Red area represents parts of the attraction basin of the attractor C , while blue parts of the attractionbasin of C . Yellow points are attracted by X ∗ which, for q > q ∗ is attractive. Points from blackarea tend to infinity. Circled regions reveal the complicated shape of attraction basins; (b) Orbit from q = (0 . , . X ∗ , even q is within unstable area, asdefined by Theorem 2 (for clarity only the first 100 iterations are shown). eferences [1] H. N. Agiza. Explicit stability zones for cournot game with 3 and 4 competitors. Chaos, Solitons and Fractals , 9(12):1955–1966, 1998.[2] E. Ahmed and H. N. Agiza. Dynamics of a cournot game with n-competitors.
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