Unstable solutions of nonautonomous linear differential equations

The fact that the eigenvalues of the family of matrices A(t) do not determine the stability of nonautonomous differential equations x' = A(t)x is well known. This point is often illustrated using examples in which the matrices A(t) have constant eigenvalues with negative real part, but the solutions of the corresponding differential equation grow in time. Here we provide an intuitive, geometric explanation of the idea that underlies these examples. The discussion is accompanied by a number of animations and easily modifiable Mathematica programs. We conclude with a discussion of possible extensions of the ideas that may provide suitable topics for undergraduate research.