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

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.