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Compound matrices and ordinary differential equations. (English) Zbl 0725.34049
A survey is given of a connection between compound matrices and ordinary differential equations. A typical result for linear systems is the following. If the n-th order differential equation x ' =A(t)x is uniformly stable, then a necessary and sufficient condition that the equation has an (n-k+1)-dimensional set of solutions satisfying lim t x(t)=0 is that y ' =A [k] (t)y should be asymptotically stable. For nonlinear autonomous systems, a criterion for orbital asymptotic stability of a closed trajectory given by Poincaré in two dimensions is extended to systems of any finite dimension. A criterion of Bendixson for the nonexistence of periodic solutions of a two dimensional system is also extended to higher dimensions.
Reviewer: P.Smith (Keele)

MSC:
34D05Asymptotic stability of ODE
34C25Periodic solutions of ODE