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

@@ -22,5 +22,22 @@ $$\text{dist}[x, c] = \min_{y \in C}|x-y|$$
 Where c is a curve. 
 Where in the plane we're using the minimum of the 2 norm.
 
+## Summary
 Stable half-paths can be generally stable, approaching an equilibrium, or periodic.
-Unstable half-paths exceed the bound $\epsilon$ somewhere.
+Unstable half-paths exceed the bound $\epsilon$ somewhere.
+
+Poincare cannot handle the time dependency of systems. As a result, we can't really use Poincare to handle real systems. That leads us to.....
+
+# Lyapunov Stability
+Basically extend the 2D distance formula we talked about last time to include n dimensions. (May need to analyze complex solutions as well).
+
+Let's define Lyapunov Stability:
+>[!note] Lyapunov Stability Definition
+>Let $x^*$ be a real or complex solution of $x = X(x,t)$. Then, 
+>1. $x^*$ is lyapunov stable iff for each value of $\epsilon>0$ however small there is a corresponding value of $\delta>0$ such that 
+>![[Pasted image 20241028152704.png]]
+>2. If the system is autonomous, then we can disregard the idea of $t_0$ in 1. 
+>3. Otherwise, we call the system unstable in the sense of Lyapunov.
+
+This stability definition defines that for an autonomous system, Lyapunov stability is sufficient for Poincare stability.
+