Instability in linear cooperative systems of ordinary differential equations

Abstract

It is well known that, contrary to the autonomous case, the stability/instability of solutions of nonautonomous linear ordinary differential equations $x^{\prime }=A(t)x$ 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.

25 pages. The author's final version (before copyediting) of a paper published in SIAM Review, with some supplementary files (mainly Wolfram Notebook files and animations) added

References in corpus (1)

• The Magnus expansion and some of its applications

Cited by in corpus (2)

• Turing instability and pattern formation on directed networks
• Turing conditions for pattern forming systems on evolving manifolds