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

Fabio Tramontana, Laura Gardini, Noemi Schmitt, Iryna Sushko, Frank Westerhoff

Research output: Contribution to journalArticle

Abstract

We reconsider the well-known 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. A simplified set of conditions determine the boundary of the stability region and we 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. We give the explicit expressions of the eigenvalues at each point of the border of the stability region in the parameter space. The results are applied to a system representing a housing market model that gives rise to a Neimark–Sacker bifurcation, a flip bifurcation or a pitchfork bifurcation.
Original languageEnglish
Pages (from-to)557-578
Number of pages22
JournalJournal of Difference Equations and Applications
Volume27
DOIs
Publication statusPublished - 2021

Keywords

  • three-dimensional maps
  • stability conditions
  • Roots of a cubic polynomial
  • necessary and sufficient conditions
  • nonlinear economic dynamics
  • codimension-1 bifurcations

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