Dichotomies and asymptotic behaviour for linear differential systems

Muldowney, James S.

doi:10.1090/S0002-9947-1984-0737880-1

Dichotomies and asymptotic behaviour for linear differential systems
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Trans. Amer. Math. Soc. 283 (1984), 465-484 Request permission

Abstract:

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.

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