Nonlinear stability question

I am looking for a simple example where a system is linearly unstable, but nonlinearly unstable or stable, depending on the sign of the initial perturbation.

For instance, assume the linear normal mode perturbation to some system goes like $a_0 e^{\lambda t}$ for $a_0$ a constant and $\lambda>0$ and real. If I then numerically integrate this and find that the full nonlinear behavior depends on the sign of $a_0$, how do I proceed in the nonlinear stability analysis? For instance, do I then assume $a_0 = \epsilon a_0(\epsilon t)$, for $\epsilon$ small and develop an equation for $a_0(\epsilon t)$ up to some asymptotic order $O(\epsilon^n)$, $n>2$, in the full nonlinear equations? I can't seem to find a canonical way to proceed.

• Nick it depends on the system. Are you looking at some type of water wave or atmospheric wave? What is the free energy source? Commented Jul 6, 2017 at 14:34
• I thought that the nonlinear stability of a system was the same as the linear stability of the system, according to the Hartman-Grobman theorem. Commented Jul 6, 2017 at 15:25
• @honeste_vivere I would love to have a simple closed system like some kind of nonlinear oscillator. The end goal is related to a problem in water waves. Commented Jul 6, 2017 at 17:36
• @probably_someone - Nope. I know of at least two systems in plasmas where they are linearly stable but nonlinearly unstable. Commented Jul 6, 2017 at 19:03
• @honeste_vivere Which systems are those? I want to look them up. Commented Jul 6, 2017 at 19:03