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<p>In this work, we investigate a two-dimensional reduction of a generic three-dimensional map, derived from a recently proposed three-firm model in discrete oligopoly theory, and analyze its local and global dynamical properties. Focusing first on the symmetric case, the map exhibits rich dynamic behavior. The Nash equilibrium loses stability via a supercritical period-doubling bifurcation. At realistic values of the parameters we prove that the 2-cycle loses stability via a Neimark–Sacker bifurcation leading to 2-cyclic closed attracting curves, that also follow a route to chaos via local and global bifurcations. These are numerically observed and the structure of the related basins of attraction is also described. Homoclinic bifurcations of the Nash equilibrium are detected as well as the homoclinic bifurcations of the fixed points on the boundaries. In the asymmetric case, similar bifurcation sequences occur giving rise to asymmetric attractors and basin geometries. Here also, local and global bifurcations are identified. These results underscore the intrinsic instability of Cournot equilibria under bounded rationality and provide new insights into the onset of chaos in oligopolistic markets.</p>

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Complex nonlinear dynamics in a two-dimensional Cournot map: from stability to chaos via local and global bifurcation scenarios

  • 期刊论文

  • 发表日期：

    2026年2月26日

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  • Mourad Azioune,
  • Mohammed Salah Abdelouahab,
  • Laura Gardini

摘要

In this work, we investigate a two-dimensional reduction of a generic three-dimensional map, derived from a recently proposed three-firm model in discrete oligopoly theory, and analyze its local and global dynamical properties. Focusing first on the symmetric case, the map exhibits rich dynamic behavior. The Nash equilibrium loses stability via a supercritical period-doubling bifurcation. At realistic values of the parameters we prove that the 2-cycle loses stability via a Neimark–Sacker bifurcation leading to 2-cyclic closed attracting curves, that also follow a route to chaos via local and global bifurcations. These are numerically observed and the structure of the related basins of attraction is also described. Homoclinic bifurcations of the Nash equilibrium are detected as well as the homoclinic bifurcations of the fixed points on the boundaries. In the asymmetric case, similar bifurcation sequences occur giving rise to asymmetric attractors and basin geometries. Here also, local and global bifurcations are identified. These results underscore the intrinsic instability of Cournot equilibria under bounded rationality and provide new insights into the onset of chaos in oligopolistic markets.