Linear Vector Fields

Abstract

Our reasons for studying linear differential equations are twofold. Firstly, whenever we wish to analyze the local phase portrait of a vector field in the neighbourhood of a singular point it is natural to linearize the problem, so that we are then investigating the phase portrait of a linear vector field (for which, incidentally, the local study at the origin and the global study are the same thing). In this chapter we shall see that such an approach does not work without considerable difficulties and that the original situation is not as close to the linear approximation as we might naively expect. Nevertheless, studying the behaviour of the integral flows of linear vector fields in some detail is justifiable, if only to extract from them those properties which have some chance of being preserved under perturbations. This leads us once more into the ideas of stability and genericity that we have already met in connection with other questions.