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arXiv · 1412.7070

Instability in linear cooperative systems of ordinary differential equations

Janusz Mierczyński
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Abstract

It is well known that, contrary to the autonomous case, the stability/instability of solutions of nonautonomous linear ordinary differential equations $x' = A(t) x$ is in no relation to the sign of the real parts of the eigenvalues of $A(t)$. In particular, the real parts of all eigenvalues can be negative and bounded away from zero, nonetheless there is a solution of magnitude growing to infinity. In this paper we present a method of constructing examples of such systems when the matrices $A(t)$ have positive off-diagonal entries (strongly cooperative systems). We illustrate those examples both with interactive animations and analytically. The paper is written in such a way that it can be accessible to students with diverse mathematical backgrounds/skills.

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Janusz Mierczyński. 2014-12-22. Instability in linear cooperative systems of ordinary differential equations. https://doi.org/10.1137/141001147

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