Instability in linear cooperative systems of ordinary differential equations

TL;DR

This paper explores the instability phenomena in nonautonomous linear cooperative systems of ODEs, showing that negative eigenvalues do not guarantee stability, and provides methods to construct such unstable systems with positive off-diagonal entries.

Contribution

It introduces a method to construct examples of unstable cooperative systems where eigenvalues suggest stability, highlighting the complex behavior of nonautonomous systems.

Findings

Examples of unstable systems with negative eigenvalues

Construction method for cooperative systems with positive off-diagonal entries

Accessible presentation with animations and analytical explanations

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.