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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 $\lim\sb{t\to \infty}x(t)=0$ is that $y'=A\sp{[k]}(t)y$ 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)

34D05Asymptotic stability of ODE
34C25Periodic solutions of ODE
Full Text: DOI
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