Bifurcation and chaos in a fractional-order Cournot duopoly game model on scale-free networks. (English) Zbl 07916730
Summary: In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter \(v_1\). The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter \(c_s\). Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.
MSC:
| 37N40 | Dynamical systems in optimization and economics |
| 39A28 | Bifurcation theory for difference equations |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional derivative; piecewise constant argument; stability; flip bifurcation; scale-free network; Cournot duopoly gameReferences:
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