We're talking all about stability

>[!note] Autonomous vs. Nonautonomous Systems
>**Autonomous**: $\dot x = X(x)$
>**Non-Autonomous:** $\dot x = X(x,t)$

We talk about stability usually meaning that things settle to an equilibrium point. But this isn't the only way to look at things...

# Poincare Stability (Path Stability)
For autonomous systems. Basically, adhere to a path for disturbances.
## Types of Paths
### Standard Path
$x^*$ is a phase path or equilibrium point whose stability is in question.
This is a solution of $\dot x = X$
### 'Half-path' or 'Half-orbit' or 'Semi-orbit'
1. Start on $a^*$ and travel on half-path $\mathcal{H}^*$ 
2. $x^*(t_0) = a^*$
$x^*$ is **Poincare stable** if all sufficiently small disturbances of the initial value $a^*$ lead to half-paths that remain a small distance from $\mathcal{H}^*$.