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Necessary and sufficient conditions for the roots of a cubic polynomial and bifurcations of codimension-1, -2, -3 for 3D maps

Author

Listed:
  • Laura Gardini

    (Department of Economics, Society & Politics, Università di Urbino Carlo Bo)

  • Noemi Schmitt

    (University of Bamberg, Bamberg, Germany,)

  • Iryna Sushko

    (Institute of Mathematics, NASU, and Kyiv School of Economics, Kyiv, Ukraine)

  • Fabio Tramontana

    (Department DiMSEFA, Università Cattolica del Sacro Cuore, Milano, Italy)

  • Frank Westerhoff

    (University of Bamberg, Bamberg, Germany)

Abstract

We reconsider the well-known Schur/Samuelson conditions, which guarantee the roots of a third-degree polynomial to be inside the unit circle. These conditions are important in the stability analysis of equilibria and cycles of three-dimensional systems in discrete time. We derive a simplified set of conditions that determine the boundary of the stability region and prove which kind of bifurcation occurs when the boundary is crossed at any of its points. These points correspond to the existence of one, two or three eigenvalues equal to 1 in modulus, real or complex conjugate, and all the remaining eigenvalues are also explicitly given. The results are applied to a system representing a housing market model that gives rise to a Neimark-Sacker bifurcation, a fliip bifurcation or a pitchfork bifurcation.

Suggested Citation

  • Laura Gardini & Noemi Schmitt & Iryna Sushko & Fabio Tramontana & Frank Westerhoff, 2019. "Necessary and sufficient conditions for the roots of a cubic polynomial and bifurcations of codimension-1, -2, -3 for 3D maps," Working Papers 1908, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2019.
    • Handle: RePEc:urb:wpaper:19_08
    as

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    File URL: http://www.econ.uniurb.it/RePEc/urb/wpaper/WP_19_08.pdf
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    References listed on IDEAS

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    More about this item

    Keywords

    Roots of a cubic polynomial; Necessary and su¢ cient conditions; Nonlinear Economic Dynamics; Stability conditions; Three-dimensional maps; Codimension-1; -2; -3 bifurcations.;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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