Periodicity Induced by Production Constraints in Cournot Duopoly Models with Unimodal Reaction Curves

  • Gian-Italo Bischi
  • Laura Gardini
  • Iryna Sushko


In the Cournot duopoly game with unimodal piecewise-linear reaction functions (tent maps) proposed by Rand (J Math Econ, 5:173–184, 1978) to show the occurrence of robust chaotic dynamics, a maximum production constraint is imposed in order to explore its effects on the long run dynamics. The presence of such constraint causes the replacement of chaotic dynamics with asymptotic periodic behaviour, characterized by fast convergence to superstable cycles. The creation of new periodic patters, as well as the possible coexistence of several stable cycles, each with its own basin of attraction, are described in terms of border collision bifurcations, a kind of global bifurcation recently introduced in the literature on non-smooth dynamical systems. These bifurcations, caused by the presence of maximum production constraint, give rise to quite particular bifurcation structures. Hence the duopoly model with constraints proposed in this paper can be seen as a simple exemplary case for the exploration of the properties of piecewise smooth dynamical systems.


Oligopoly games Dynamical systems Constraints Border collision bifurcations 

JEL classification

C61 C73 L13 



This work is developed in the framework of the research project on “Dynamic Models for behavioural economics” financed by DESP-University of Urbino.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • Gian-Italo Bischi
    • 1
  • Laura Gardini
    • 1
  • Iryna Sushko
    • 2
  1. 1.DESP-Department of Economics, Society, PoliticsUniversity of UrbinoUrbinoItaly
  2. 2.Institute of Mathematics NASUKievUkraine

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