Topological structure of the phase portrait for linear vector fields in R3 and applications - Silva, Luan Lima da; Tonon, Durval José (C.Q.D. - Revista Eletrônica Paulista de Matemática, 2025)
This paper presents an analysis of the sign of the real part of the roots 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 problem has already been treated in the literature without significant advances. As an application of the results obtained, we will fully characterize the dynamics and topological structure of the phase portrait for linear fields in R3. In addition, we present applications in the local qualitative study of equilibrium points for vector fields in R3 defined with one or more parameters.