Dichotomies and Asymptotic Behaviour for Linear Differential Systems

Sufficient conditions that a system of differential equations x' = A{l)x have a dichotomy usually require that the matrix 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). An asymptotic analysis is also given for subspaces of the solution set by means of the associated compound equations.