Overstability

Overstability is oscillatory instability: a perturbation grows while continuing to change sign or direction periodically. A real perturbation can contain $e^{\sigma t}(a\cos \omega t+b\sin \omega t)$ with $\sigma >0$ and $\omega \ne 0$, corresponding to a complex-conjugate pair of linear growth rates with positive real part.

The oscillation period is approximately $2\pi /|\omega |$, while $1/\sigma$ sets the envelope-growth time. Successive peak heights can therefore reveal growth even when a short segment of the trace resembles an ordinary oscillation. This differs from a stationary instability with a purely real positive growth rate and from a damped oscillation with a negative envelope growth rate. The distinction is useful when classifying how an equilibrium first becomes unstable as a control parameter changes, including an oscillatory onset associated with a Hopf bifurcation.