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Peer-reviewedCC BY 4.0Axioms

Nonlinear Dynamics, Bifurcation, Basin Analysis, and Chaos Control in an Asymmetric-Cost Bertrand Duopoly Model

  • Mourad Azioune1
  • Messaoud Berkal1,2
  • Mohammed-Salah Abdelouahab1,2
  • René Lozi3
  • Mohammed Bakheet Almatrafi4
4 affiliations
  1. 1Laboratory of Mathematics and Their Interactions, Department of Mathematics, Abdelhafid Boussouf University, Mila 43000, Algeria
  2. 2Department of Applied Mathematics, Abdelhafid Boussouf University, R.P. 26, Mila 43000, Algeria
  3. 3Laboratoire Jean Alexandre Dieudonné, CNRS, Université Côte d’Azur, Campus Sciences, Parc Valrose, 28 Avenue Valrose, 06108 Nice Cedex 02, France
  4. 4Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah 30002, Saudi Arabia
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Abstract

This paper investigates a dynamic Bertrand duopoly model with boundedly rational firms under asymmetric cost structures. Unlike classical formulations based on symmetric costs, we consider a heterogeneous market where one firm faces a quadratic cost function and the other a linear cost structure. We analyze the existence and local stability of the Bertrand–Nash equilibrium, revealing how adjustment speeds and cost asymmetry influence market stability. The results show that excessive price adjustments and increasing cost differences can destabilize the equilibrium through a period-doubling route to chaos, generating unpredictable price dynamics. The chaotic behavior is characterized using bifurcation analysis, Lyapunov exponents, basin of attraction analysis, and the 0–1 chaos test. Furthermore, a feedback control strategy is developed to suppress chaotic fluctuations and restore the Nash equilibrium. By integrating asymmetric cost structures, global stability analysis, chaos detection, and control mechanisms within a Bertrand framework, this study extends existing symmetric-cost models and provides new insights into stabilizing competitive markets. Numerical simulations confirm the theoretical findings and highlight the role of cost heterogeneity and adjustment mechanisms in shaping the stability and predictability of price competition.

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Cite

BibTeXRIS
BibTeX
@article{azioune2026nonlinear,
  title = {{Nonlinear Dynamics, Bifurcation, Basin Analysis, and Chaos Control in an Asymmetric-Cost Bertrand Duopoly Model}},
  author = {Mourad Azioune and Messaoud Berkal and Mohammed-Salah Abdelouahab and René Lozi and Mohammed Bakheet Almatrafi},
  journal = {Axioms},
  year = {2026},
  month = sep,
  doi = {10.3390/axioms15100725},
  url = {https://www.mdpi.com/2075-1680/15/10/725},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Nonlinear Dynamics, Bifurcation, Basin Analysis, and Chaos Control in an Asymmetric-Cost Bertrand Duopoly Model
AU  - Mourad Azioune
AU  - Messaoud Berkal
AU  - Mohammed-Salah Abdelouahab
AU  - René Lozi
AU  - Mohammed Bakheet Almatrafi
PY  - 2026
DA  - 2026/09/30
JO  - Axioms
DO  - 10.3390/axioms15100725
UR  - https://www.mdpi.com/2075-1680/15/10/725
AB  - This paper investigates a dynamic Bertrand duopoly model with boundedly rational firms under asymmetric cost structures. Unlike classical formulations based on symmetric costs, we consider a heterogeneous market where one firm faces a quadratic cost function and the other a linear cost structure. We analyze the existence and local stability of the Bertrand–Nash equilibrium, revealing how adjustment speeds and cost asymmetry influence market stability. The results show that excessive price adjustments and increasing cost differences can destabilize the equilibrium through a period-doubling route to chaos, generating unpredictable price dynamics. The chaotic behavior is characterized using bifurcation analysis, Lyapunov exponents, basin of attraction analysis, and the 0–1 chaos test. Furthermore, a feedback control strategy is developed to suppress chaotic fluctuations and restore the Nash equilibrium. By integrating asymmetric cost structures, global stability analysis, chaos detection, and control mechanisms within a Bertrand framework, this study extends existing symmetric-cost models and provides new insights into stabilizing competitive markets. Numerical simulations confirm the theoretical findings and highlight the role of cost heterogeneity and adjustment mechanisms in shaping the stability and predictability of price competition.
SN  - 2075-1680
ER  -