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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 $
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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)
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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. $
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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)
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Figure 5.
Chaotic attractor of
the proposed game with $\nu =0.98$ and for $\alpha =2,%
\varepsilon _{1}=1.44,\varepsilon _{2}=1.1,\varepsilon _{3}=1.1,\zeta_{1}=0.4,\zeta_{2}=0.8,\zeta_{3}=0.1,\gamma _{1}=0.07,
\gamma _{2}=0.03,\gamma _{3}=0.4$.
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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 $
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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 $
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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. $
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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 $
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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 $
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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 $
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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 $
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Figure 13.
ApEn of the game model (8) vesrus $ \nu$
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