From faebd5d307023551824fc556f23f8fae3b5c771d Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 18 Nov 2024 13:30:02 -0500 Subject: [PATCH] vault backup: 2024-11-18 13:30:02 --- .../2024-11-18 Volume Contraction.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) 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 c4c49449..d2ec5efa 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 @@ -1,7 +1,7 @@ Lorenz system is dissapative. This means: - Volume in phase space contracts with flow? This introduces some questions... How do volumes evolve? - +n Suppose a surface $S(t)$ encloses volume $V(t)$, with normal vectors pointing away from the surface ($\vec{n}$). A trajectory starts on S. let them evolve for $dt$. With a flux vector $\vec{f}$, we have @@ -18,4 +18,4 @@ 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 +Proving which type something will end up on is much harder. But, repellers will always result in a positive $\dot V$. \ No newline at end of file