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<p>In this study, we investigate a discretized fractional-order Cournot duopoly model. To discretize the model, the piecewise constant argument approach is applied. We study the existence and the stability at fixed points of the discretized fractional-order Cournot duopoly model. In addition, we fully investigate the existence and direction of period-doubling and Neimark–Sacker bifurcations at the positive fixed point using center manifold and bifurcation theory. We apply feedback control and hybrid control techniques to reduce chaos and bifurcation. Numerical examples confirm our theoretical results and reveal the model’s complex dynamics. Notably, we demonstrate that increasing the speed adjustment parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4882_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(k_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> in the first competitor’s mechanism can drive the system from a stable equilibrium through periodic oscillations to chaos, underscoring the critical role of this parameter in market complexity and instability.</p>

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Dynamic complexity in a discretized fractional-order Cournot duopoly model using piecewise constant argument method

  • 期刊论文

  • 发表日期：

    2025年8月8日

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  • Rizwan Ahmed,
  • Muhammad Adnan,
  • Abdul Rauf

摘要

In this study, we investigate a discretized fractional-order Cournot duopoly model. To discretize the model, the piecewise constant argument approach is applied. We study the existence and the stability at fixed points of the discretized fractional-order Cournot duopoly model. In addition, we fully investigate the existence and direction of period-doubling and Neimark–Sacker bifurcations at the positive fixed point using center manifold and bifurcation theory. We apply feedback control and hybrid control techniques to reduce chaos and bifurcation. Numerical examples confirm our theoretical results and reveal the model’s complex dynamics. Notably, we demonstrate that increasing the speed adjustment parameter \(k_1\) k 1 in the first competitor’s mechanism can drive the system from a stable equilibrium through periodic oscillations to chaos, underscoring the critical role of this parameter in market complexity and instability.