The objective of the article is to study the dynamics of the proposed fractional-order Cournot triopoly game. Sufficient conditions for the existence and uniqueness of the triopoly game solution are obtained. Stability analysis of equilibrium points of the fractional-order game is also discussed. The conditions for the presence of Nash equilibrium point along with its global stability analysis are studied. The interesting dynamical behaviors of the arbitrary-order Cournot triopoly game are discussed. Moreover, the effects of seasonal periodic forcing on the game’s behaviors are examined. The 0–1 test is used to distinguish between regular and irregular dynamics of system behaviors. Numerical analysis is used to verify the theoretical results that are obtained, and revealed that the nonautonomous fractional-order model induces more complicated dynamics in the Cournot triopoly game behavior and the seasonally forced game exhibits more complex dynamics than the unforced one.
AbdelazizMAIsmailAIAbdullahFA, et al. (2018) Bifurcations and chaos in a discrete SI epidemic model with fractional order. Advances in Difference Equations2018(1): 44.
2.
AghababaMP (2014a) Chaotic behavior in fractional-order horizontal platform systems and its suppression using a fractional finite-time control strategy. Journal of Mechanical Science and Technology28: 1875–1880.
3.
AghababaMP (2014b) Fractional modeling and control of a complex nonlinear energy supply-demand system. Complexity20(6): 74–86.
4.
AghababaMPBorjkhaniM (2014) Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme. Complexity20(2): 37–46.
5.
AgizaHNElsadanyAA (2004) Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Applied Mathematics and Computation149: 843–860.
6.
AhmedEAgizaHNHassanSZ (2000) On modifications of Puu’s dynamical duopoly. Chaos, Solitons & Fractals11: 1025–1028.
7.
AhmedEElgazzarAS (2007) On fractional order differential equations model for nonlocal epidemics. Physica A: Statistical Mechanics and its Applications379: 607–614.
8.
AhmedEEl-SayedAMAEl-SakaHAA (2007) Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. Journal of Mathematical Analysis and Applications325: 542–553.
9.
AndaluzJElsadanyAAJarneG (2017) Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations. Mathematics and Computers in Simulation132: 86–99.
10.
AndaluzJJarneG (2015) On the dynamics of economic games based on product differentiation. Mathematics and Computers in Simulation113: 16–27.
11.
AskarS (2014) The rise of complex phenomena in Cournot duopoly games due to demand functions without inflection points. Communications in Nonlinear Science and Numerical Simulation19: 1918–1925.
12.
AskarSSAl-khedhairiA (2017) Analysis of nonlinear duopoly games with product differentiation: stability, global dynamics, and control. Discrete Dynamics in Nature and Society2017: 1–13.
13.
BensonDAMeerschaertMMRevielleJ (2013) Fractional calculus in hydrologic modeling: a numerical perspective. Advances in Water Resources51: 479–497.
14.
BischiGNaimzadaA (1999) Global analysis of a duopoly game with bounded rationality. Advances in Dynamic Games and Applications5: 361–385.
15.
BischiGIChiarellaCKopelM, et al. (2010) Nonlinear Oligopolies: Stability and Bifurcations. New York: Springer.
16.
CournotA (1963) Researches into the Principles of the Theory of Wealth (Engl. Trans.). Irwin Paper Back Classics in Economics. Hachette: Paris.
17.
DiethelmK (1997) An algorithm for the numerical solution of differential equations of fractional order. Electronic Transactions on Numerical Analysis5: 1–6.
18.
DiethelmKFordNJ (2002) Analysis of fractional differential equations. Journal of Mathematical Analysis265: 229–248.
19.
DiethelmKFordNJFreedAD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics29: 3–22.
20.
Dubiel-TeleszynskiT (2011) Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale. Communications in Nonlinear Science and Numerical Simulation16: 296–308.
21.
El-MisieryAEMAhmedE (2006) On a fractional model for earthquakes. Applied Mathematics and Computation178: 207–211.
22.
ElsadanyAAAwadAM (2016) Dynamical analysis of a delayed monopoly game with a log-concave demand function. Operations Research Letters44: 33–38.
23.
ElsadanyAAMatoukAE (2014) Dynamic Cournot duopoly game with delay. Journal of Complex Systems2014: 1–7.
24.
ElsadanyAAMatoukAE (2015) Dynamical behaviors of fractional-order Lotka-Volterra predator–prey model and its discretization. Journal of Applied Mathematics and Computing49: 269–283.
25.
El-SayedAMA (1996) Fractional-order diffusion-wave equation. International Journal of Theoretical Physics35: 311–322.
26.
GhazianiRAlidoustiJEshkaftakiAB (2016) Stability and dynamics of a fractional order Leslie-Gower prey-predator model. Applied Mathematical Modelling40: 2075–2086.
27.
GottwaldGAMelbourneI (2004) A new test for chaos in deterministic systems. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences460(2042): 603–611.
28.
GottwaldGAMelbourneI (2005) Testing for chaos in deterministic systems with noise. Physica D: Nonlinear Phenomena212: 100–110.
29.
GottwaldGAMelbourneI (2009) On the Implementation of the 0–1 Test for Chaos. Society for Industrial and Applied Mathematics Journal on Applied Dynamical Systems8: 129–145.
30.
GottwaldGAMelbourneI (2016) The 0–1 test for chaos: a review. In: SkokosC, et al. (eds) Chaos Detection and Predictability. Lecture Notes in Physics. Berlin, Heidelberg: Springer, 221–247.
31.
HommesC (2013) Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems. Cambridge: Cambridge University Press.
32.
HuangCCaoJXiaoM, et al. (2017) Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders. Applied Mathematics and Computation293: 293–310.
33.
LiCPengG (2004) Chaos in Chen’s system with a fractional order. Chaos, Solitons & Fractals22: 443–450.
34.
LiH-LZhangLHuC, et al. (2017a) Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing54: 435–449.
35.
LiZChenDMaM, et al. (2017b) Feigenbaum’s constants in reverse bifurcation of fractional-order Rössler system. Chaos, Solitons & Fractals99: 116–123.
36.
MaJSiF (2016) Complex dynamics of a continuous Bertrand Duopoly game model with two-stage delay. Entropy18(7): 266.
37.
MachadoJTMachadoJTVMainardiF (2011) Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation16: 1140–1153.
38.
MatignonD (1996) Stability result for fractional differential equations with applications to control processing. In: MACS-SMC Proceedings, Lille, France, July 1996, pp. 963–968. USA: Computational Engineering in Systems Applications.
39.
MatoukAEElsadanyAA (2014) Achieving synchronization between the fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique. Applied Mathematics Letters29: 30–35.
40.
MatoukAEElsadanyAA (2016) Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model. Nonlinear Dynamics85: 1597–1612.
41.
MatoukAEElsadanyAAAhmedE, et al. (2015) Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization. Communications in Nonlinear Science and Numerical Simulation27: 153–167.
42.
MatoukAEElsadanyAAXinB (2017) Neimark-Sacker bifurcation analysis and complex nonlinear dynamics in a heterogeneous quadropoly game with an isoelastic demand function. Nonlinear Dynamics89(4): 2533–2552.
43.
MeralFCRoystonTJMaginR (2010) Fractional calculus in viscoelasticity: an experimental study. Communications in Nonlinear Science and Numerical Simulation15: 939–945.
PuuT (1996) Complex dynamics with three oligopolists. Chaos, Solitons & Fractals12: 207–581.
46.
PuuT (2008) On the stability of Cournot equilibrium when the number of competitors increases. Journal of Economic Behavior & Organization66: 445–456.
47.
SaporaACornettiPCarpinteriA (2013) Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach. Communications in Nonlinear Science and Numerical Simulation18: 63–74.
48.
SnyderBCVan GorderRAVajraveluK (2013) Continuous-time dynamic games for the Cournot adjustment process for competing oligopolists. Applied Mathematics and Computation219: 6400–6409.
49.
SunZMaJ (2012) Complexity of triopoly price game in Chinese cold rolled steel market. Nonlinear Dynamics67: 2001–2008.
50.
TavazoeiMSHaeriM (2008) Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Analysis: Theory, Methods & Applications. 69: 1299–1320.
51.
Tenreiro MachadoJAMataME (2015) Pseudo phase plane and fractional calculus modeling of western global economic downturn. Communications in Nonlinear Science and Numerical Simulation22: 396–406.
52.
TramontanaFElsadanyAEA (2012) Heterogeneous triopoly game with isoelastic demand function. Nonlinear Dynamics68: 187–193.
53.
ZhangX (2017) Dynamics of a class of fractional-order nonautonomous Lorenz-type systems. Chaos: An Interdisciplinary Journal of Nonlinear Science27: 041104.
