Dichotomies and asymptotic behaviour for linear differential systems

Abstract

Sufficient conditions that a system of differential equations x ′ = A ( t ) x x’ = A(t)x have a dichotomy usually require that the matrix A ( t ) A(t) be bounded or at least that some restriction be placed on the rate of growth or decay of solutions. Here three sets of necessary and sufficient conditions for a dichotomy which do not impose such a restriction are given in terms of Liapunov functions. Each of the theorems gives practical criteria for a dichotomy including the extension to unbounded matrices of criteria which depend on a concept of diagonal dominance for A ( t ) A(t) . An asymptotic analysis is also given for subspaces of the solution set by means of the associated compound equations.