Topological structure of the phase portrait for linear vector fields in R3 and applications

Abstract

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.