Stability analysis of delayed cournot model in the sense of Lyapunov

Abstract

In this paper the dynamic equation of Duopoly production model in certain two firms is considered. The existence of best responses that can maximize profit, and stability conditions are analyzed when one of the players or both of them have delayed information and/or delayed actions. A system of nonlinear delayed differential equations and Lyapunov method of nonlinear stability analysis are employed. It is ascertained that, in the case of equal and fixed information delay in both the firms, the delay causes oscillatory process in the system and does not affect the qualitative behavior of the solution (no effect on the stability of the Nash equilibrium point), but only changes the transition process. On the other hand, when one of the firms has implementation delay and the rival player makes decision without delay, it leads to instability of the dynamic system at least locally. The same result is obtained when one of the firms has implementation and the other information delay. Numerical simulation using MATLAB2012a is used to demonstrate the applicability and accuracy of the results.

Keywords: Delay Differential Equations, Nonlinear dynamic system, Stability, Lyapunov method, method of linearization, Nash equilibrium, duopoly model