Border collision bifurcations in a piecewise linear duopoly model

We consider a duopoly model characterized by a two-dimensional non-invertible continuous map T given by piecewise linear functions, with several partitions, defining a duopoly game. The structure of the game is such that it has separate second iterate so that its dynamics can be studied via a one-dimensional composite function, that is piecewise linear with multiple partitions in which the definition of the map changes. The number of partitions may change from 2 to 5, depending on the parameters. The dynamics are characterized by degenerate bifurcations and border collision bifurcations, which are typical in maps having kink points. Here the peculiarity is the multiplicity of the partitions, which leads to bifurcations different from those occurring in maps with only one kink point. We show several bifurcations, coexistence of cycles, attracting and superstable, as well chaotic attractors and chaotic repellors, related to the outcome of particular border collision bifurcations.