Chapter ?? discussed three methods that are used to solve planar systems of linear, constant coefficient, ordinary differential equations. The last method is based on similarity and the explicit computation of the matrix exponential for certain normal form matrices. This method depends crucially on the classification of matrices up to similarity given in Chapter ??, Theorem ??.

In this chapter we explore qualitative features of phase portraits for planar linear systems of differential equations using similarity. We find that the qualitative theory is completely determined by the eigenvalues and eigenvectors of the coefficient matrix — which is not surprising given that we can classify matrices up to similarity by just knowing their eigenvalues and eigenvectors. The set of planar phase portraits divides systems of linear differential equations into two camps: hyperbolic and nonhyperbolic. The hyperbolic systems consist of saddles, sinks and sources, while the nonzero nonhyperbolic systems consist of centers, saddle nodes, and shears.