From ec088ca6f7c13e4e1170881efdcaddbbfdea90b8 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 9 Sep 2024 13:01:00 -0400 Subject: [PATCH] vault backup: 2024-09-09 13:01:00 --- .../2024-09-09 Frameworks and Review.md | 14 ++++++++++++++ 1 file changed, 14 insertions(+) diff --git a/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md b/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md index dfb753755..0e867731b 100644 --- a/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md +++ b/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 + +