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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 \(\lim_{t\to \infty}x(t)=0\) 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:

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

### Keywords:

compound matrices; ordinary differential equations; orbital asymptotic stability; Poincaré; criterion of Bendixson; nonexistence of periodic solutions
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\textit{J. S. Muldowney}, Rocky Mt. J. Math. 20, No. 4, 857--872 (1990; Zbl 0725.34049)

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