From 177a43a14063803909cec1b0c4420636d3e38748 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 18 Nov 2024 13:23:41 -0500 Subject: [PATCH] vault backup: 2024-11-18 13:23:41 --- .../2024-11-18 Volume Contraction.md | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md index a791910d9..c4c494493 100644 --- a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md +++ b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md @@ -13,3 +13,9 @@ $$V(t+dt) = V(t) + \int_S (\vec f \cdot \vec n dt)dA $$ $$\dot{V} = \int_S (\vec f \cdot \vec n)dA $$ Now we can apply the divergence theorem: $$\dot{V} = \int_V (\nabla \cdot \vec f )dV $$ +If you start with a solid blob of initial conditions, this integral will evaluate down to where things end up. If $\dot V$ is negative, then the system will converge to a stable subspace. +Limiting set will consist of +- fixed points +- limit cycles +- strong attractors +Proving which type something will end up on \ No newline at end of file