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

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.

Keywords:

• Roots of a cubic polynomial
• necessary and sufficient conditions
• nonlinear economic dynamics
• stability conditions
• three-dimensional maps
• codimension-1
• -2
• -3 bifurcations

2010 Mathematics Subject Classifications:

• 37C25
• 37C75
• 37C05
• 37G10
• 39A60

Acknowledgments

Gardini, Sushko and Tramontana conducted this study within the research project on Models of Behavioural Economics for Sustainable Development, at the Department of Economics, Society, Politics (DESP) of the University of Urbino.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Recall that for a polynomial $T(x)=x^2+a_{1}x+a_2$ the well known stability conditions by Jury [27] are given by $T(1)>0$, $T(-1)>0,$ $|a_2|<1$.