Topological structure of the phase portrait for linear vector fields in R3 and applications (C.Q.D. - Revista Eletrônica Paulista de Matemática, 2025)
This paper presents an analysis of the sign of the real part of theroots of a polynomial of degree 3 as a function of its coefficients.The problem is treated mainly from an analytical point of view,knowing that from an algebraic point of view the problemhas already been treated in the literature without significantadvances. As an application of the results obtained, we willfully characterize the dynamics and topological structure of thephase portrait for linear fields in R3. In addition, we presentapplications in the local qualitative study of equilibrium pointsfor vector fields in R3 defined with one or more parameters.
Keywords: Tridimensional linear vector fields; Roots of cubic polynomials; Local dynamics near to equilibrium points; Bifurcation diagram