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Zbl 0725.34049
Muldowney, James S.
Compound matrices and ordinary differential equations. (English)
[J] Rocky Mt. J. Math. 20, No.4, 857-872 (1990). ISSN 0035-7596

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.
[P.Smith (Keele)]

MSC 2000:: *34D05 Asymptotic stability of ODE
34C25 Periodic solutions of ODE

Keywords: compound matrices; ordinary differential equations; orbital asymptotic stability; Poincaré; criterion of Bendixson; nonexistence of periodic solutions

Cited in: Zbl 1042.34075 Zbl 0943.91047

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