Complex dynamics investigations of a mixed Bertrand duopoly game: synchronization and global analysis

Nonlinear Dynamics - In this paper, an economic competition between two firms that want to maximize the weighted-average social welfare and own profits is proposed. This kind of competition is...     

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.     

Dynamical analysis and chaos control in a heterogeneous Kopel duopoly game

A dynamic of a nonlinear Kopel duopoly game with heterogeneous players is presented. By assuming two heterogeneous players where one player use naive expectation whereas the other employs a technique of adaptive. The stability conditions of equilibrium points are analyzed. Numerical simulations are used to show bifurcation diagrams, phase portraits and sensitive dependence on initial conditions. The chaotic behavior of the game has been controlled by using feedback control method.     

Competition analysis of a triopoly game with bounded rationality

A dynamic Cournot game characterized by three boundedly rational players is modeled by three nonlinear difference equations. The stability of the equilibria of the discrete dynamical system is analyzed. As some parameters of the model are varied, the stability of Nash equilibrium is lost and a complex chaotic behavior occurs. Numerical simulation results show that complex dynamics, such as, bifurcations and chaos are displayed when the value of speed of adjustment is high. The global complexity analysis can help players to take some measures and avoid the collapse of the output dynamic competition game.     

Chaotic dynamics of a discrete prey–predator model with Holling type II

A discrete-time prey–predator model with Holling type II is investigated. For this model, the existence and stability of three fixed points are analyzed. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for different parameters of the model. The fractal dimension of a strange attractor of the model was also calculated. Numerical simulations show that the discrete model exhibits rich dynamics compared with the continuous model, which means that the present model is a chaotic, and complex one.     

Dynamics and chaos control of a duopolistic Bertrand competitions under environmental taxes

Annals of Operations Research - This paper investigates the difference between price and quantity competition in a mixed duopoly game. We describe the behavior of a duopolistic Bertrand competition...     

Heterogeneous triopoly game with isoelastic demand function

In this paper, we analyze a triopolistic market with heterogeneous firms when the demand function is isoelastic. We consider the same heterogeneous firms as Elabbasy et al. (Comput. Math. Appl. 57:488?C499, 2009) introducing a nonlinearity in the demand function instead of the cost function. Stability conditions of the two equilibrium points and complex dynamics are studied. The main novelty consists of the double route to chaos, via period-doubling bifurcations and via Neimark?CSacker bifurcation. The two routes have important differences from the economic point of view.     

Neimark–Sacker bifurcation analysis and complex nonlinear dynamics in a heterogeneous quadropoly game with an isoelastic demand function

In this paper, a nonlinear quadropoly game based on Cournot model with fully heterogeneous players is established. This game extends the model introduced by Tramontana and Elsadany (Nonlinear Dyn 68:187–193, 2012) who considered a heterogeneous triopoly game with an isoelastic demand function. Here, four different types of players and potentially different marginal costs are considered. Moreover, the assumption of an isoelastic demand function increases the nonlinearity of the final four-dimensional map. The stability of the resulting discrete-time dynamical system is analyzed. The existence of Neimark–Sacker bifurcation near the Nash equilibrium point of the game is shown. Also, based on the Kuznetsov’s normal form technique for discrete-time system, the stability of the Neimark–Sacker bifurcation is also discussed which indicates that the bifurcation is supercritical. Moreover, it is shown that the Nash equilibrium point of the game undergoes period-doubling (flip) bifurcation. Furthermore, the double route to chaotic dynamics in this model, via flip bifurcations and via Neimark–Sacker bifurcation of the Nash equilibrium point, is illustrated. Coexistence of multi-chaotic attractors of the model is illustrated. Simulation tools like bifurcation diagrams, stability regions of parameters, Lyapunov exponent spectrum, phase plots and basins of attraction are used to verify the complex dynamics of the game.