Linear Systems

Abstract

In this chapter, we undertake a detailed investigation of the rather special class of planar autonomous differential equations where the vector field is given by a linear map. By exploiting special properties of solutions of linear systems, with a small dose of linear algebra, we will be able to compute the flows of these systems explicitly and determine their phase portraits. After obtaining explicit solutions, we direct our attention to qualitative questions and classify linear systems up to flow equivalence. We also investigate certain bifurcation phenomena within the class of linear systems. We conclude the chapter with several useful facts about the solutions of nonautonomous linear differential equations. Admittedly, striving for explicit solutions may seem somewhat of a deviation from our earlier efforts. However, this is one of the few situations where such a complete answer, possessing its own mathematical appeal, exists. Furthermore, this information will be important in the local qualitative analysis of equilibrium points of nonlinear systems. The success of obtaining explicit solutions of linear systems is not, however, without an ironic disappointment. The important task of deciding the qualitative equivalence of two linear systems requires considerably more mathematics than mere formulas for their explicit solutions, as we shall see in this chapter.