Stability of an ODE

An ODE is stable if, given any $ϵ>0$, there exists a $\delta (ϵ)>0$ such that for two solutions $y_A(t)$ and $y_B(t)$ with $|y_A(t_0)-y_B(t_0)|\le \delta$:

$$|y_A(t)-y_B(t)|\le ϵ\text{for all }t>t_0$$

So a stable ODE is one where small differences in initial conditions are not amplified.

Unstable ODE: $\dot{y}=y$

The solution is $y(t)=y_0{e}^t$. For two solutions with slightly different initial conditions:

$$|y_A(t)-y_B(t)|=|y_{0,A}-y_{0,B}|e^t\to \infty$$

There is no finite upper bound, so no valid $\delta$ can be found - this ODE is unstable.

Stable ODE: $\dot{y}=-y$

The solution is $y(t)=y_0{e}^{-t}$. For two solutions with slightly different initial conditions:

$$|y_A(t)-y_B(t)|=|y_{0,A}-y_{0,B}|e^{-t}\to 0$$

This goes to zero as $t\to \infty$, so we can simply take $\delta =ϵ$. The initial difference is attenuated (damped), and hence this ODE is stable.