The phase space structure of the oligopoly dynamical system by means of Darboux integrability

TL;DR

This paper analyzes the phase space structure of a three-firm Cournot oligopoly model using Darboux integrability, revealing invariant manifolds, stability properties, and the role of first integrals in understanding system dynamics.

Contribution

It introduces the application of Darboux polynomial methods to study integrability and phase space structure in oligopoly models, providing new insights into their dynamical behavior.

Findings

Phase space contains invariant manifolds for monopoly and duopoly.

Unique stable node acts as a global attractor in the positive quadrant.

First integrals help reduce system dimension and analyze trajectory behavior.

Abstract

We investigate the dynamical complexity of Cournot oligopoly dynamics of three firms by using the qualitative methods of dynamical systems to study the phase structure of this model. The phase space is organized with one-dimensional and two-dimensional invariant submanifolds (for the monopoly and duopoly) and unique stable node (global attractor) in the positive quadrant of the phase space (Cournot equilibrium). We also study the integrability of the system. We demonstrate the effectiveness of the method of the Darboux polynomials in searching for first integrals of the oligopoly. The general method as well as examples of adopting this method are presented. We study Darboux non-integrability of the oligopoly for linear demand functions and find first integrals of this system for special classes of the system, in particular, rational integrals can be found for a quite general set of model parameters. We show how first integral can be useful in lowering the dimension of the system using the example of $n$ almost identical firms. This first integral also gives information about the structure of the phase space and the behaviour of trajectories in the neighbourhood of a Nash equilibrium