Border collision bifurcations in a piecewise linear duopoly model

Abstract

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.

Keywords:

• Two-dimensional duopoly maps
• piecewise linear maps
• border collision bifurcations
• superstable cycles
• degenerate bifurcations

Mathematics Subject Classifications:

• 39A23
• 39A28
• 39A30
• 39A33
• 39A60

Acknowledgments

The work of both authors has been done within the activities of the research project on ‘Dynamic models in economics and finance’ of the Department DESP of the University of Urbino.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Davide Radi acknowledges the financial support from the Grantová Agentura České Republiky (Czech Scientific Foundation) under project 23-06282S and from the VŠB–TU Ostrava under the SGS research project SP2023/19. This work was supported by Grantová Agentura České Republiky.