Dichotomies and Asymptotic Behaviour for Linear Differential Systems

Sufficient conditions that a system of differential equations $x’ = A(t)x$ 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.