Linear Differential Equations

Abstract

In this chapter we obtain corresponding results to those in Chapter 3 for continuous time. More precisely, we study in detail the class of Lyapunov exponents defined by the solutions of a nonautonomous linear equation. In particular, we obtain lower and upper bounds for the Grobman coefficient. Again these bounds are very useful for nonregular equations. The lower bound is established for an arbitrary coefficient matrix, whereas the upper bound is obtained for an upper-triangular coefficient matrix. On the other hand, we also show that from the point of view of the theory of regularity, one can always reduce the study of a linear equation to the study of one with an upper-triangular coefficient matrix. We then show that the notion of regularity can be characterized in terms of exponential growth rates of volumes and we establish various important properties of regular equations. Finally, we consider the stronger notion of regularity for a two-sided coefficient matrix.