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

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+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 distrubances.
+## 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'