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.

我们重新考虑保证三次多项式的根在单位圆内的众所周知的条件。这些条件在离散时间三维系统的平衡和循环的稳定性分析中很重要。一组简化的条件确定了稳定区域的边界，我们证明了当边界在其任何点处交叉时会发生哪种分叉。这些点对应于模数、实数或复共轭中存在一个、两个或三个等于 1 的特征值。我们给出了参数空间中稳定区域边界的每个点的特征值的显式表达式。将结果应用于代表住房市场模型的系统，该模型产生 Neimark-Sacker 分叉、翻转分叉或干草叉分叉。