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Local bifurcations of three and four-dimensional systems : A tractable characterization with economic applications

Author

Listed:
  • Stefano Bosi

    (EPEE - Centre d'Etudes des Politiques Economiques - UEVE - Université d'Évry-Val-d'Essonne)

  • David Desmarchelier

    (BETA - Bureau d'Économie Théorique et Appliquée - INRA - Institut National de la Recherche Agronomique - UNISTRA - Université de Strasbourg - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique)

Abstract

We provide necessary and sufficient conditions to detect local bifurcations of three and four-dimensional dynamical systems in continuous time. We characterize not only the bifurcations of codimension one but also those of codimension two. For the sake of completeness, we give also the non-degeneracy conditions for each bifurcation. The added value of our methodology rests on its generality. To illustrate the tractability of our approach, we provide two analytical applications of dimension three and four to environmental economics, complemented with numerical simulations.

Suggested Citation

  • Stefano Bosi & David Desmarchelier, 2019. "Local bifurcations of three and four-dimensional systems : A tractable characterization with economic applications," Post-Print hal-02512900, HAL.
    • Handle: RePEc:hal:journl:hal-02512900
    DOI: 10.1016/j.mathsocsci.2018.11.001
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    Cited by:

    1. Abeer Hamdan Alblowy & Normah Maan & Sana Abdulkream Alharbi, 2022. "Role of Glucose Risk Factors on Human Breast Cancer: A Nonlinear Dynamical Model Evaluation," Mathematics, MDPI, vol. 10(19), pages 1-20, October.
    2. Guerrazzi, Marco, 2024. "Wage bargaining and capital accumulation: A dynamic version of the monopoly union model," MPRA Paper 121433, University Library of Munich, Germany.
    3. Maxime Menuet & Alexandru Minea & Patrick Villieu & Anastasios Xepapadeas, 2024. "Environmental quality along the process of economic growth: a theoretical reappraisal," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 77(4), pages 1219-1258, June.
    4. Bosi, Stefano & Desmarchelier, David, 2018. "An economic model of metapopulation dynamics," Ecological Modelling, Elsevier, vol. 387(C), pages 196-204.
    5. Ogochukwu Ifeacho & Gilberto González-Parra, 2025. "Impact of Delayed Decaying Corruption Effects on a Socioeconomic System with Economic Growth and Unemployment," Mathematics, MDPI, vol. 13(11), pages 1-28, May.
    6. Guillaume MOREL, 2020. "A note on pollution and infectious disease," Working Papers of BETA 2020-38, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    7. Ogochukwu Ifeacho & Gilberto González-Parra, 2025. "Mathematical Modeling of Economic Growth, Corruption, Employment and Inflation," Mathematics, MDPI, vol. 13(7), pages 1-32, March.
    8. Stefano Bosi & David Desmarchelier, 2021. "Pollution effects on disease transmission and economic stability," International Journal of Economic Theory, The International Society for Economic Theory, vol. 17(2), pages 169-189, June.
    9. David Desmarchelier & Rémi Girard, 2023. "Renewable resource and harvesting cost in a simple monetary overlapping generation economy," Revue d'économie industrielle, De Boeck Université, vol. 0(4), pages 147-171.
    10. Estelle CAMPENET & David DESMARCHELIER, 2025. "A note on pollution inertia and endogenous cycles in Ramsey economies," Working Papers of BETA 2025-10, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    11. Guillaume MOREL, 2020. "PA note on pollution and infectious diseases," Working Papers of BETA 2020-22, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    12. Guillaume Morel, 2020. "A note on pollution and infectious disease," Economics Bulletin, AccessEcon, vol. 40(4), pages 2723-2733.
    13. David Desmarchelier & Magali Jaoul‐Grammare & Guillaume Morel & Thi K. C. Pham, 2025. "Infectious disease and endogenous cycles: Lockdown hits two birds with one stone," International Journal of Economic Theory, The International Society for Economic Theory, vol. 21(3), pages 316-342, September.
    14. Marco Guerrazzi & Giuseppe Candido, 2025. "The determination of the price of capital goods: A differential game approach," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 46(1), pages 222-234, January.
    15. Maxime MENUET & Alexandru MINEA & Patrick VILLIEU & Anastasios XEPAPADEAS, 2021. "Growth, Endogenous Environmental Cycles, and Indeterminacy," LEO Working Papers / DR LEO 2889, Orleans Economics Laboratory / Laboratoire d'Economie d'Orleans (LEO), University of Orleans.
    16. Bella, Giovanni & Liuzzi, Danilo & Mattana, Paolo & Venturi, Beatrice, 2020. "Equilibrium selection in an environmental growth model with a S-shaped production function," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    17. Guerrazzi, Marco, 2025. "A two-sector model of optimal growth in which labour is employed only in the industry of investment goods: A complete characterization of equilibrium paths," MPRA Paper 126299, University Library of Munich, Germany.
    18. Maxime Menuet & Alexandru Minea & Patrick Villieu & Anastasios Xepapadeas, 2020. "Economic Growth and the Environment: A Theoretical Reappraisal," DEOS Working Papers 2031, Athens University of Economics and Business.

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    Keywords

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles
    • O44 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - Environment and Growth

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