The following image depicts a linear, nonautonomous system y' = A(t) y,
which is presented in
Instability in Linear Cooperative Systems of Ordinary Differential Equations.
The matrix A(t) is periodic and piecewise constant alternating between
two values with the same, real, negative spectrum. However, the norm
of the solution does not decay.
Animation by Yash Chandak, Jennifer Li, Autumn Phaneuf, Toby Schwed, and Angelica Simonetti