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Local Bifurcations of Three and Four-Dimensional Systems: A Tractable Characterization with Economic Applications

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  • Stefano Bosi
  • David Desmarchelier

Abstract

We provide necessary and sufficient conditions to detect local bifurcations of three and four-dimensional dynamical systems in continuous time. We characterize the bifurcations of codimension one and two. Our methodology is both general and tractable. To illustrate its operability, we provide two analytical applications of dimension three and four to environmental economics, complemented with numerical simulations.

Suggested Citation

  • Stefano Bosi & David Desmarchelier, 2017. "Local Bifurcations of Three and Four-Dimensional Systems: A Tractable Characterization with Economic Applications," EUSP Department of Economics Working Paper Series 2017/02, European University at St. Petersburg, Department of Economics.
    • Handle: RePEc:eus:wpaper:ec2017_02
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    References listed on IDEAS

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    1. Stefano Bosi & David Desmarchelier, 2017. "A simple method to study local bifurcations of three and four-dimensional systems: characterizations and economic applications," Working Papers 2017.16, FAERE - French Association of Environmental and Resource Economists.
    2. Ayong Le Kama, Alain D., 2001. "Sustainable growth, renewable resources and pollution," Journal of Economic Dynamics and Control, Elsevier, vol. 25(12), pages 1911-1918, December.
    3. Stefano Bosi & David Desmarchelier, 2018. "Limit Cycles Under a Negative Effect of Pollution on Consumption Demand: The Role of an Environmental Kuznets Curve," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 69(2), pages 343-363, February.
    4. Barnett, William A. & He, Susan, 2010. "Existence of singularity bifurcation in an Euler-equations model of the United States economy: Grandmont was right," Economic Modelling, Elsevier, vol. 27(6), pages 1345-1354, November.
    5. Jean-Paul Barinci & Jean-Pierre Drugeon, 2017. "Assessing the Local Stability Properties of Discrete Three-Dimensional Dynamical Systems: A Geometrical Approach with Triangles and Planes and an Application with Some Cones," Studies in Economic Theory, in: Kazuo Nishimura & Alain Venditti & Nicholas C. Yannelis (ed.), Sunspots and Non-Linear Dynamics, chapter 0, pages 15-39, Springer.
    6. Bosi, Stefano & Desmarchelier, David, 2018. "Natural cycles and pollution," Mathematical Social Sciences, Elsevier, vol. 96(C), pages 10-20.
    7. Wirl, Franz, 2004. "Sustainable growth, renewable resources and pollution: Thresholds and cycles," Journal of Economic Dynamics and Control, Elsevier, vol. 28(6), pages 1149-1157, March.
    8. Grandmont, Jean-Michel, 1985. "On Endogenous Competitive Business Cycles," Econometrica, Econometric Society, vol. 53(5), pages 995-1045, September.
    9. Jean-Pierre Drugeon & Jean-Paul Barinci, 2017. "Assessing the Local Stability Properties of Discrete Three-Dimensional Dynamical Systems: A Geometrical Approach with Triangles and Planes and an Application with Some Cones," Post-Print halshs-01884337, HAL.
    10. Stefano Bosi & David Desmarchelier, 2017. "Are the Laffer curve and the green paradox mutually exclusive?," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 19(5), pages 937-956, October.
    11. Grandmont, Jean-Michel, 2008. "Nonlinear difference equations, bifurcations and chaos: An introduction," Research in Economics, Elsevier, vol. 62(3), pages 122-177, September.
    12. Engelbert Dockner & Gustav Feichtinger, 1991. "On the optimality of limit cycles in dynamic economic systems," Journal of Economics, Springer, vol. 53(1), pages 31-50, February.
    13. Grandmont, Jean-Michel & Pintus, Patrick & de Vilder, Robin, 1998. "Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles," Journal of Economic Theory, Elsevier, vol. 80(1), pages 14-59, May.
    14. Kazuo Nishimura & Alain Venditti & Nicholas C. Yannelis (ed.), 2017. "Sunspots and Non-Linear Dynamics," Studies in Economic Theory, Springer, number 978-3-319-44076-7, March.
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    Cited by:

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    2. 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.
    3. David DESMARCHELIER & Rémi GIRARD, 2022. "Renewable resource and harvesting cost in a simple monetary overlapping generation economy," Working Papers of BETA 2022-32, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    4. Guillaume MOREL, 2020. "PA note on pollution and infectious diseases," Working Papers of BETA 2020-22, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    5. Bosi, Stefano & Desmarchelier, David, 2018. "An economic model of metapopulation dynamics," Ecological Modelling, Elsevier, vol. 387(C), pages 196-204.
    6. Guillaume Morel, 2020. "A note on pollution and infectious disease," Economics Bulletin, AccessEcon, vol. 40(4), pages 2723-2733.
    7. David Desmarchelier & Magali Jaoul-Grammare & Guillaume Morel & Thi Kim Cuong Pham, 2021. "Infectious disease and endogenous cycles: lockdown hits two birds with one stone," Working Papers of BETA 2021-23, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    8. Guerrazzi, Marco & Candido, Giuseppe, 2023. "The determination of the price of capital goods: A differential game approach," MPRA Paper 119118, University Library of Munich, Germany.
    9. 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.
    10. 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).
    11. 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.
    12. Guillaume MOREL, 2020. "A note on pollution and infectious disease," Working Papers of BETA 2020-38, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.

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    More about this item

    Keywords

    Local bifurcations; Codimensions one and two; Pollution; Natural capital;
    All these keywords.

    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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