And the idea is also applicable to trajectories rather than singular points: if you change your starting point by an epsilon, would the trajectory be vastly similar or a bit different? or very different? cue in Lyapunov fun and Lipschitz continuity[0] as a metric and, to a lesser extent, conditions for chaotic trajectories to emerge.
In some systems you want to generate constant oscillations, so your system can be in stable perpetual dynamic equilibrium producing a nice sine wave.
Along these terms, I think equilibrium doesn't really make sense - it means stability in many definitions. From the top google result:
> Equilibrium is defined as a state of balance or a stable situation where opposing forces cancel each other out and where no changes are occurring. An example of equilibrium is in economics when supply and demand are equal. An example of equilibrium is when you are calm and steady.
Another take on this: (Stable vs unstable equilibrium) https://web.ma.utexas.edu/users/davis/375/popecol/lec9/equil....