vault backup: 2024-10-28 15:17:14

300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md

@@ -15,4 +15,12 @@ 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 $\mathcal{H}^*$.
+$x^*$ is **Poincare stable** if all sufficiently small disturbances of the initial value $a^*$ lead to half-paths that remain a small distance from $\mathcal{H}^*$.
+![[Pasted image 20241028151006.png]]![[Pasted image 20241028151117.png]]
+## How do we define distances?
+$$\text{dist}[x, c] = \min_{y \in C}|x-y|$$
+Where c is a curve. 
+Where in the plane we're using the minimum of the 2 norm.
+
+Stable half-paths can be generally stable, approaching an equilibrium, or periodic.
+Unstable half-paths exceed the bound $\epsilon$ somewhere.