# American Institute of Mathematical Sciences

• Previous Article
On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type
• DCDS-B Home
• This Issue
• Next Article
Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains

## The effect of caputo fractional difference operator on a novel game theory model

 1 Laboratory of Dynamical Systems and Control, University of Larbi Ben M'hidi, Oum El Bouaghi, Algeria 2 Department of Mathematics, Faculty of Science, University of Ha'il, Kingdom of Saudi Arabia 3 Department of Mathematics and Computer Science, University of Larbi Ben M'hidi, Oum El Bouaghi, Algeria 4 Mathematics Department, Faculty of Science, University of Ha'il, Kingdom of Saudi Arabia 5 Universita del Salento, Dipartimento Ingegneria Innovazione, 73100 Lecce, Italy 6 Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author: phamvietthanh@tdtu.edu.vn

Received  May 2020 Revised  August 2020 Published  October 2020

It is well-known that fractional-order discrete-time systems have a major advantage over their integer-order counterparts, because they can better describe the memory characteristics and the historical dependence of the underlying physical phenomenon. This paper presents a novel fractional-order triopoly game with bounded rationality, where three firms producing differentiated products compete over a common market. The proposed game theory model consists of three fractional-order difference equations and is characterized by eight equilibria, including the Nash fixed point. When suitable values for the fractional order are considered, the stability of the Nash equilibrium is lost via a Neimark-Sacker bifurcation or via a flip bifurcation. As a consequence, a number of chaotic attractors appear in the system dynamics, indicating that the behaviour of the economic model becomes unpredictable, independently of the actions of the considered firm. The presence of chaos is confirmed via both the computation of the maximum Lyapunov exponent and the 0-1 test. Finally, an entropy algorithm is used to measure the complexity of the proposed game theory model.

Citation: Amina-Aicha Khennaoui, A. Othman Almatroud, Adel Ouannas, M. Mossa Al-sawalha, Giuseppe Grassi, Viet-Thanh Pham. The effect of caputo fractional difference operator on a novel game theory model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020302
##### References:
 [1] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.  Google Scholar [2] A. Al-khedhairi, Differentiated Cournot duopoly game with fractional-order and its discretization, Engineering Computations, 36 (2019), 26. Google Scholar [3] A. Al-Khedhairi, Dynamics of a Cournot duopoly game with a generalized bounded rationality, Complexity, 2020 (2020), 8903183. Google Scholar [4] G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model., 52 (2010), 556-566.  doi: 10.1016/j.mcm.2010.03.055.  Google Scholar [5] S. S. Askar and A. Al-Khedhairi, Analysis of a four-firm competition based on a generalized bounded rationality and different mechanisms, Complexity, 2019 (2019), 6352796. doi: 10.1155/2019/6352796.  Google Scholar [6] F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I, 3 (2009), 1-12.  doi: 10.14232/ejqtde.2009.4.3.  Google Scholar [7] H. M. Baskonus, T. Mekkaoui, Z. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.  doi: 10.3390/e17064255.  Google Scholar [8] J. Cermak, I. Gyori and L. Nechvatal, On explicit stability conditions for a linear fractional difference system, Fractional Calculus and Applied Analysis, 18 (2015), 651-672. doi: 10.1515/fca-2015-0040.  Google Scholar [9] G. A. Gottwald and I. Melbourne, On the implementation of the 0-1 test for chaos, SIAM J. Appl. Dyn. Syst., 8 (2009), 129-145.  doi: 10.1137/080718851.  Google Scholar [10] N. Laskin, Fractional market dynamics, Physica A: Statist. Mech. Appl., 287 (2000), 482-492.  doi: 10.1016/S0378-4371(00)00387-3.  Google Scholar [11] Y. Li, C. Sun, H. Ling, A. Lu and Y. Liu, Oligopolies price game in fractional order system, Chaos, Solitons & Fractals, 132 (2020), 109583. doi: 10.1016/j.chaos.2019.109583.  Google Scholar [12] S. M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA., 88 (1991), 2297–2301. doi: 10.1073/pnas.88.6.2297.  Google Scholar [13] F. Sapuppo, M. Bucolo, M. Intaglietta, L. Fortuna and P. Arena, A cellular nonlinear network: Real-time technology for the analysis of microfluidic phenomena in blood vessels, Nanotechnology, 17 (2006), S54. doi: 10.1088/0957-4484/17/4/009.  Google Scholar [14] T. $\breve{S}$kovránek, I. Podlubny and I. Petrá$\breve{s}$, Modeling of the national economies in state-space: A fractional calculus approach, Economic Modelling, 29 (2012), 1322-1327.   Google Scholar [15] H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlin. Sci. Numer. Simul., 64 (2018), 213-231.  doi: 10.1016/j.cnsns.2018.04.019.  Google Scholar [16] I. Tejado, D. Valério and N. Valério, Fractional calculus in economic growth modeling. The Portuguese case, in ICFDA'14 International Conference on Fractional Differentiation and Its Applications, IEEE, 2014, 1–6. Google Scholar [17] B. Xin, W. Peng and Y. Kwon, A fractional-order difference Cournot duopoly game with long memory, preprint, arXiv: 1903.04305. doi: 10.1016/j.physa.2020.124993.  Google Scholar [18] B. Xin and Y. Li, 0-1 test for chaos in a fractional order financial system with investment incentive, Abstract and Applied Analysis, 2013 (2013), 876298. doi: 10.1155/2013/876298.  Google Scholar [19] B. Xin, W. Peng and L. Guerrini, A continuous time Bertrand duopoly game with fractional delay and conformable derivative: Modelling, discretization process, Hopf bifurcation and chaos, Frontiers in Physics, 7 (2019), 84. Google Scholar [20] C. Xu, M. Liao, P. Li, Q. Xiao and S. Yuan, Control strategy for a fractional-order chaotic financial model, Complexity, 2019 (2019), 2989204. doi: 10.1155/2019/2989204.  Google Scholar [21] A. Yousefpour, H. Jahanshahi, J. M. Munoz-Pacheco, S. Bekiros and Z. Wei, A fractional-order hyper-chaotic economic system with transient chaos, Chaos, Solitons & Fractals, 130 (2020), 109400. doi: 10.1016/j.chaos.2019.109400.  Google Scholar

show all references

##### References:
 [1] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.  Google Scholar [2] A. Al-khedhairi, Differentiated Cournot duopoly game with fractional-order and its discretization, Engineering Computations, 36 (2019), 26. Google Scholar [3] A. Al-Khedhairi, Dynamics of a Cournot duopoly game with a generalized bounded rationality, Complexity, 2020 (2020), 8903183. Google Scholar [4] G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model., 52 (2010), 556-566.  doi: 10.1016/j.mcm.2010.03.055.  Google Scholar [5] S. S. Askar and A. Al-Khedhairi, Analysis of a four-firm competition based on a generalized bounded rationality and different mechanisms, Complexity, 2019 (2019), 6352796. doi: 10.1155/2019/6352796.  Google Scholar [6] F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I, 3 (2009), 1-12.  doi: 10.14232/ejqtde.2009.4.3.  Google Scholar [7] H. M. Baskonus, T. Mekkaoui, Z. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.  doi: 10.3390/e17064255.  Google Scholar [8] J. Cermak, I. Gyori and L. Nechvatal, On explicit stability conditions for a linear fractional difference system, Fractional Calculus and Applied Analysis, 18 (2015), 651-672. doi: 10.1515/fca-2015-0040.  Google Scholar [9] G. A. Gottwald and I. Melbourne, On the implementation of the 0-1 test for chaos, SIAM J. Appl. Dyn. Syst., 8 (2009), 129-145.  doi: 10.1137/080718851.  Google Scholar [10] N. Laskin, Fractional market dynamics, Physica A: Statist. Mech. Appl., 287 (2000), 482-492.  doi: 10.1016/S0378-4371(00)00387-3.  Google Scholar [11] Y. Li, C. Sun, H. Ling, A. Lu and Y. Liu, Oligopolies price game in fractional order system, Chaos, Solitons & Fractals, 132 (2020), 109583. doi: 10.1016/j.chaos.2019.109583.  Google Scholar [12] S. M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA., 88 (1991), 2297–2301. doi: 10.1073/pnas.88.6.2297.  Google Scholar [13] F. Sapuppo, M. Bucolo, M. Intaglietta, L. Fortuna and P. Arena, A cellular nonlinear network: Real-time technology for the analysis of microfluidic phenomena in blood vessels, Nanotechnology, 17 (2006), S54. doi: 10.1088/0957-4484/17/4/009.  Google Scholar [14] T. $\breve{S}$kovránek, I. Podlubny and I. Petrá$\breve{s}$, Modeling of the national economies in state-space: A fractional calculus approach, Economic Modelling, 29 (2012), 1322-1327.   Google Scholar [15] H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlin. Sci. Numer. Simul., 64 (2018), 213-231.  doi: 10.1016/j.cnsns.2018.04.019.  Google Scholar [16] I. Tejado, D. Valério and N. Valério, Fractional calculus in economic growth modeling. The Portuguese case, in ICFDA'14 International Conference on Fractional Differentiation and Its Applications, IEEE, 2014, 1–6. Google Scholar [17] B. Xin, W. Peng and Y. Kwon, A fractional-order difference Cournot duopoly game with long memory, preprint, arXiv: 1903.04305. doi: 10.1016/j.physa.2020.124993.  Google Scholar [18] B. Xin and Y. Li, 0-1 test for chaos in a fractional order financial system with investment incentive, Abstract and Applied Analysis, 2013 (2013), 876298. doi: 10.1155/2013/876298.  Google Scholar [19] B. Xin, W. Peng and L. Guerrini, A continuous time Bertrand duopoly game with fractional delay and conformable derivative: Modelling, discretization process, Hopf bifurcation and chaos, Frontiers in Physics, 7 (2019), 84. Google Scholar [20] C. Xu, M. Liao, P. Li, Q. Xiao and S. Yuan, Control strategy for a fractional-order chaotic financial model, Complexity, 2019 (2019), 2989204. doi: 10.1155/2019/2989204.  Google Scholar [21] A. Yousefpour, H. Jahanshahi, J. M. Munoz-Pacheco, S. Bekiros and Z. Wei, A fractional-order hyper-chaotic economic system with transient chaos, Chaos, Solitons & Fractals, 130 (2020), 109400. doi: 10.1016/j.chaos.2019.109400.  Google Scholar
The phase portraits of game (8) with parameter values $\alpha = 2, \, \varepsilon_1 = 1.042811791, \, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ for different fractional order values: (a) $\nu = 1, (b)\, \nu = 0.9, (c) \, \nu = 0.865, (d)\, \nu = 0.81$
(a) Bifurcation diagram versus $\nu$ when $\alpha = 2, \varepsilon _{1} = 1.042811791, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$.(b) The maximum Lyapunov exponents with respect to $\nu$ corresponding to (a)
(a) Bifurcation diagram versus $\varepsilon _1$ with order $\nu = 0.985$ when $\alpha = 2, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$. (b) Bifurcation diagram versus $\varepsilon _{1}$ with order $\nu = 0.972.$
(a) Bifurcation diagram versus $\nu$ when $\alpha = 2, \varepsilon _{1} = 1.44, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$. (b) The maximum Lyapunov exponents with respect to $\nu$ corresponding to (a)
Chaotic attractor of the proposed game with $\nu =0.98$ and for $\alpha =2,% \varepsilon _{1}=1.44,\varepsilon _{2}=1.1,\varepsilon _{3}=1.1,\zeta_{1}=0.4,\zeta_{2}=0.8,\zeta_{3}=0.1,\gamma _{1}=0.07, \gamma _{2}=0.03,\gamma _{3}=0.4$.
Periodic attractor of the proposed game with $\nu = 0.975$ and for $\alpha = 2, \varepsilon _{1} = 1.44,\varepsilon _{2} = 1.1,\varepsilon _{3} = 1.1,\zeta_{1} = 0.4,\zeta_{2} = 0.8,\zeta_{3} = 0.1,\gamma _{1} = 0.07, \gamma _{2} = 0.03,\gamma _{3} = 0.4$
Chaotic attractor of the proposed game with $\nu = 0.96$ and for $\alpha = 2, \varepsilon _{1} = 1.44, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$
(a) Bifurcation diagram versus $\varepsilon_1$ with order $\nu = 1$ when $\alpha = 1, \varepsilon _{2} = 0.9, \varepsilon _{3} = 0.9, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ (b) Bifurcation diagram versus $\varepsilon _{1}$ with order $\nu = 0.7635.$
Periodic attractor of the proposed game with $\nu = 0.7635$ for $\varepsilon_1 = 2.6$ and $\alpha = 1, \varepsilon _{2} = 0.9,\varepsilon _{3} = 0.9,\zeta_{1} = 0.4,\zeta_{2} = 0.8,\zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$
Chaotic attractor of the proposed game with $\nu = 0.7635$ for $\varepsilon_1 = 2.9$ and $\alpha = 1, \varepsilon _{2} = 0.9,\varepsilon _{3} = 0.9,\zeta_{1} = 0.4,\zeta_{2} = 0.8,\zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$
0-1 test: regular dynamics of the translation components $(p, q)$ of the Cournot game (8) for $\alpha = 2$, $\varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ with fractional order $\nu = 0.865$
0-1 test: regular dynamics of the translation components $(p, q)$ of the Cournot game (8) for $\alpha = 2, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ with fractional order $\nu = 0.7635$
ApEn of the game model (8) vesrus $\nu$
 [1] Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144 [2] Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 [3] Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177 [4] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003 [5] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [6] Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353 [7] Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166 [8] Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003 [9] Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024 [10] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [11] Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269 [12] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [13] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 [14] Guo-Niu Han, Huan Xiong. Skew doubled shifted plane partitions: Calculus and asymptotics. Electronic Research Archive, 2021, 29 (1) : 1841-1857. doi: 10.3934/era.2020094 [15] Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409 [16] Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1447-1468. doi: 10.3934/dcdsb.2020168 [17] Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021002 [18] Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459 [19] Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012 [20] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

2019 Impact Factor: 1.27