Dynamic behavior and bifurcation analysis of a Cournot duopoly model with log-linear price function and quadratic cost functions

Bashir Al-Hdaibat; A. Alameer; Bashir Al-Hdaibat; A. Alameer

doi:10.3934/math.20251214

AIMS Mathematics

2025, Volume 10, Issue 11: 27608-27634. doi: 10.3934/math.20251214

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Dynamic behavior and bifurcation analysis of a Cournot duopoly model with log-linear price function and quadratic cost functions

Bashir Al-Hdaibat ^{1
,
,},
A. Alameer ²

1.
Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa, Jordan
2.
Department of Mathematics, University of Hafr Al-Batin, Hafr Al-Batin 31991, Saudi Arabia

Received: 31 August 2025 Revised: 18 October 2025 Accepted: 28 October 2025 Published: 26 November 2025
MSC : 39A10, 39A28, 65P20, 65P30, 97N80

In this paper, we studied the dynamic behavior of a Cournot-type duopoly model with homogeneous goods, a convex log-linear price function, and quadratic cost functions. By using the Lambert $ W $ function, we derived explicit expressions for the fixed points and showed that the model has exactly three fixed points. We conducted a local stability analysis to identify the stability regions for each fixed point. Furthermore, we demonstrated that the system undergoes a sequence of period-doubling bifurcations, which give rise to stable cycles of period 2 and period 4. We showed that the period-4 cycle loses its stability through a Neimark-Sacker bifurcation, where an attracting invariant closed curve emerges. We also derived the topological normal forms associated with the period-doubling bifurcation points. To confirm the presence of chaotic behavior, we computed the largest Lyapunov exponents. Finally, numerical simulations, along with bifurcation analyses performed using MatContM, validated our theoretical results. These findings extend and enhance previous studies in the literature, particularly the results in ^[1,2].
- duopoly model,
- nonlinear dynamics,
- stability analysis,
- bifurcations,
- chaos,
- MatContM
Citation: Bashir Al-Hdaibat, A. Alameer. Dynamic behavior and bifurcation analysis of a Cournot duopoly model with log-linear price function and quadratic cost functions[J]. AIMS Mathematics, 2025, 10(11): 27608-27634. doi: 10.3934/math.20251214

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Abstract

In this paper, we studied the dynamic behavior of a Cournot-type duopoly model with homogeneous goods, a convex log-linear price function, and quadratic cost functions. By using the Lambert $ W $ function, we derived explicit expressions for the fixed points and showed that the model has exactly three fixed points. We conducted a local stability analysis to identify the stability regions for each fixed point. Furthermore, we demonstrated that the system undergoes a sequence of period-doubling bifurcations, which give rise to stable cycles of period 2 and period 4. We showed that the period-4 cycle loses its stability through a Neimark-Sacker bifurcation, where an attracting invariant closed curve emerges. We also derived the topological normal forms associated with the period-doubling bifurcation points. To confirm the presence of chaotic behavior, we computed the largest Lyapunov exponents. Finally, numerical simulations, along with bifurcation analyses performed using MatContM, validated our theoretical results. These findings extend and enhance previous studies in the literature, particularly the results in ^[1,2].

References

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