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.

Gardini, L., Schmitt, N., Sushko, I., Tramontana, F., Westerhoff, F., Necessary and sufficient conditions for the roots of a cubic polynomial and bifurcations of codimension-1, -2, -3 for 3D maps, <<JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS>>, 2021; 27 (4): 557-578. [doi:10.1080/10236198.2021.1920937] [http://hdl.handle.net/10807/181271]

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

Tramontana, F.;
2021

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.
Inglese
Gardini, L., Schmitt, N., Sushko, I., Tramontana, F., Westerhoff, F., Necessary and sufficient conditions for the roots of a cubic polynomial and bifurcations of codimension-1, -2, -3 for 3D maps, <<JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS>>, 2021; 27 (4): 557-578. [doi:10.1080/10236198.2021.1920937] [http://hdl.handle.net/10807/181271]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/181271
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