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' = A(t)x bears 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, but 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.