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A simple method to study local bifurcations of three and four-dimensional systems: characterizations and economic applications

Author

Listed:
  • Stefano Bosi

    (EPEE, University of Evry)

  • David Desmarchelier

    (BETA, University of Lorraine)

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. The added value of this methodology rests on its generality and tractability. To illustrate the simplicity 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, 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.
    • Handle: RePEc:fae:wpaper:2017.16
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    References listed on IDEAS

    as
    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 of BETA 2017-07, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    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. 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.
    5. Bosi, Stefano & Desmarchelier, David, 2018. "Natural cycles and pollution," Mathematical Social Sciences, Elsevier, vol. 96(C), pages 10-20.
    6. 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.
    7. 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.
    8. Grandmont, Jean-Michel, 2008. "Nonlinear difference equations, bifurcations and chaos: An introduction," Research in Economics, Elsevier, vol. 62(3), pages 122-177, September.
    9. 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.
    10. 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, July.
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    Cited by:

    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 of BETA 2017-07, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    2. Bosi, Stefano & Desmarchelier, David, 2019. "Local bifurcations of three and four-dimensional systems: A tractable characterization with economic applications," Mathematical Social Sciences, Elsevier, vol. 97(C), pages 38-50.
    3. Bo Sang & Bo Huang, 2020. "Zero-Hopf Bifurcations of 3D Quadratic Jerk System," Mathematics, MDPI, vol. 8(9), pages 1-19, August.

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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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