Delay Cournot Duopoly Game with Gradient Adjustment: Berezowski Transition from a Discrete Model to a Continuous Model

Akio Matsumoto,Ferenc Szidarovszky,Keiko Nakayama

doi:10.3390/math9010032

Akio Matsumoto, Ferenc Szidarovszky + Show 1 more

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https://doi.org/10.3390/math9010032

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Abstract

This paper investigates the asymptotical behavior of the equilibrium of linear classical duopolies by reconsidering the two-delay model with two different positive delays. In a two-dimensional analysis, the stability switching curves were first analytically determined. Numerical studies verified and illustrated the theoretical results. In the sensitivity analysis it was demonstrated that the inertia coefficient has a twofold effect: enlarges the stability region as well as simplifies the complicated dynamics with period-halving cascade. In contrary, the adjustment speed contracts the stability region and complicates simple dynamics with period-doubling bifurcation. In addition, for various values of τ1 and τ2, a wide variety of dynamics appears ranging from simple cycle via a Hopf bifurcation to chaotic oscillations.

Highlights

Game with Gradient Adjustment: Berezowski Transition from a Discrete ModelThe earliest studies on oligopoly theory focused on the existence and uniqueness to a Continuous Model
If the firms are in an equilibrium state, the common interest of
We examine the no-delay case of τ1 = τ2 = 0 in which the characteristic equation reads λ2 + 2αxe(σ1 + σ2) λ + 3(αxe)2 = 0

Summary

Introduction

Game with Gradient Adjustment: Berezowski Transition from a Discrete ModelThe earliest studies on oligopoly theory focused on the existence and uniqueness to a Continuous Model. In the case of continuous time scales, differential-difference equations describe the dynamic processes.

Results

Conclusion