Duopolistic Stackelberg game: investigation of complex dynamics and chaos control

  • S. S. Askar
Original Paper


The current paper introduces a duopolistic Stackelberg game under the assumption of differentiated products. The game is built based on a cost function that is nonlinear and depends on the quantities produced by firms and the announced plan products. Indeed, there is a difference between the actual product and the plan product. Here, we try to confirm that the equilibrium production of a firm is no more than its announced plan product. The dynamic of the proposed game is described by a nonlinear discrete dynamical system with bounded rationality mechanism by which the time evolution of the competing firms is analyzed. The equilibrium point of the system is obtained and its local stability is discussed using the eigenvalues properties or Jury conditions. The discussion shows that the stability of equilibrium point is affected by the speed of adjustment parameter and some other cost parameters. Confirmation of this discussion is supported by some numerical simulations that lead to the existence of complex dynamic phenomena such as bifurcation and chaos. In addition, we are applied a feedback control scheme to suppress and overcome chaos existed and at the same time to force the system to go back to its stabilization behavior.


Bounded rationality Stackelberg game Stability Bifurcation Chaos Control 



The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its Funding this Research Group No. (RG-1435-054).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Operations Researches, College of ScienceKing Saud UniversityRiyadhSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

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