A geometric classification of equilibria by spectral types

We write three algebraic equations whose variables are the principal invariants of a matrix. When the matrix is the linearization of a vector field at an equilibrium, the loci defined by such equations correspond to specific types of bifurcations, and the complement of such loci is a decomposition of the space of principal invariants in open domains in which the equilibrium has a given spectral type. When dealing with a parameter-dependent dynamical system, the three equations can be pulled back from the space of invariants to the parameter space, and provide the bifurcation decomposition of parameter space for the dynamical system. In this article, we develop and apply this technique thoroughly in dimensions 3 and 4, and we give effective methods to compute the spectral indices.