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