An algebro-geometric classification of spectral types of equilibria

Abstract

We give three algebraic equations which allow a geometric classification of all spectral types of equilibria of a given $m$-dimensional dynamical system, and we analyse them thoroughly in dimension 3 and 4. The loci defined by these equations correspond to definite types of bifurcations. The complement of such loci give a geometric decomposition of the space of invariants in open domains in which the equilibrium has a given spectral types. The usefulness of this approach lays in the fact that, when dealing with a parameter-dependent dynamical system, the pull-back of the loci from the space of invariants to the parameter space gives the bifurcation-decomposition of parameter space for the dynamical system at hand. We also give effective methods to explicitly compute the spectral indices.