American Institute of Mathematical Sciences

Advanced
August  2021, 26(8): 4549-4565. doi: 10.3934/dcdsb.2020302

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

Amina-Aicha Khennaoui 1, , A. Othman Almatroud 2, , Adel Ouannas 3, , M. Mossa Al-sawalha 4, , Giuseppe Grassi 5, and  Viet-Thanh Pham 6,,

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  August 2021 Early access  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.

Keywords: Cournot game, dynamic system, discrete fractional calculus, memory, chaos.
Mathematics Subject Classification: 37-00, 91A25, 26A33, 44A55, 65P20, 37A35.
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, 2021, 26 (8) : 4549-4565. 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]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

Figure 1.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  (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)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  (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. $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  (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)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Chaotic attractor of the proposed game with $\nu =0.98$ and for $\alpha =2,% \varepsilon _{1}=1.44,\varepsilon _{2}=1.1,\varepsilon _{3}=1.1,\zeta_{1}=0.4,\zeta_{2}=0.8,\zeta_{3}=0.1,\gamma _{1}=0.07, \gamma _{2}=0.03,\gamma _{3}=0.4$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  (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. $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 13.  ApEn of the game model (8) vesrus $ \nu$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021012
[2]

Akio Matsumoto, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021069
[3]

Tao Li, Suresh P. Sethi. A review of dynamic Stackelberg game models. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 125-159. doi: 10.3934/dcdsb.2017007
[4]

Viviana Alejandra Díaz, David Martín de Diego. Generalized variational calculus for continuous and discrete mechanical systems. Journal of Geometric Mechanics, 2018, 10 (4) : 373-410. doi: 10.3934/jgm.2018014
[5]

Ido Polak, Nicolas Privault. A stochastic newsvendor game with dynamic retail prices. Journal of Industrial & Management Optimization, 2018, 14 (2) : 731-742. doi: 10.3934/jimo.2017072
[6]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013
[7]

Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373
[8]

Jaydeep Swarnakar. Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021007
[9]

S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660
[10]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2155-2185. doi: 10.3934/cpaa.2021062
[11]

Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 539-560. doi: 10.3934/dcdss.2020030
[12]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
[13]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210
[14]

Zhujun Jing, K.Y. Chan, Dashun Xu, Hongjun Cao. Bifurcations of periodic solutions and chaos in Josephson system. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 573-592. doi: 10.3934/dcds.2001.7.573
[15]

Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263
[16]

Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259
[17]

Chrystie Burr, Laura Gardini, Ferenc Szidarovszky. Discrete time dynamic oligopolies with adjustment constraints. Journal of Dynamics & Games, 2015, 2 (1) : 65-87. doi: 10.3934/jdg.2015.2.65
[18]

Jianhong Wu, Weiguang Yao, Huaiping Zhu. Immune system memory realization in a population model. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 241-259. doi: 10.3934/dcdsb.2007.8.241
[19]

Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949
[20]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (194)
  • HTML views (310)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences