Asymptotic dynamics of a piecewise smooth map modelling a competitive market

In the present work we study asymptotic dynamics of a multi-dimensional piecewise smooth map which models an oligopoly market where competitors use adaptive scheme for reaction choice. Each competitor also defines the moment for renewing the capital equipment depending on how intensively the latter is used. Namely, the larger output is produced, the quicker the capital exhausts. It is shown then that the asymptotic dynamics of the map allows coexistence of different metric attractors in which case it is sensitive to initial conditions. We also investigate stability of trajectories representing Cournot equilibria which are here not fixed but periodic points. In particular, it is shown that several such Cournot equilibria, belonging to different invariant manifolds, may coexist some of them being locally asymptotically stable and some being unstable.