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300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md

@@ -30,3 +30,17 @@ Really our options come down to:
 - Solve an approximation to the problem
 We mix and match these approaches. 
 ## Geometric (Qualitative) Methods
+Geometric analysis answers questions like "is this stable?" "what's the response look like?"
+### Linear Systems
+$$ \dot{x} = Ax$$
+This is a simple linear dynamic system. 
+How many equilibria does this system have?
+**One.**
+The system is at equilibrium where $\frac{dx}{dt} = 0$. It won't move from this point.
+
+Is this system stable? Check the eigenvalues of A.
+### Nonlinear Systems
+**Recall: $\dot{x} = 1-2\cos x$**
+#### The Phase Plane
+
+