期刊
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS
卷 27, 期 4, 页码 557-578出版社
TAYLOR & FRANCIS LTD
DOI: 10.1080/10236198.2021.1920937
关键词
Roots of a cubic polynomial; necessary and sufficient conditions; nonlinear economic dynamics; stability conditions; three-dimensional maps; codimension-1; -2; -3 bifurcations
This study reconsiders the conditions for the roots of a third-degree polynomial to be inside the unit circle and their importance in stability analysis. A simplified set of conditions determine the boundary of the stability region and predict the type of bifurcation that will occur when the boundary is crossed. These findings are applied to a housing market model, resulting in different types of bifurcations.
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.
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