From 4a03627c6f76c6443e20a83e670ab4984392c4d4 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 28 Oct 2024 15:06:50 -0400 Subject: [PATCH] vault backup: 2024-10-28 15:06:50 --- .../2024-10-28 Stability.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md index baf8c839..d6d7d59d 100644 --- a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md +++ b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-10-28 Stability.md @@ -7,11 +7,12 @@ We're talking all about stability 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. +For autonomous systems. Basically, adhere to a path for disturbances. ## 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' 1. Start on $a^*$ and travel on half-path $\mathcal{H}^*$ -2. $ \ No newline at end of file +2. $x^*(t_0) = a^*$ +$x^*$ is **Poincare stable** if all sufficiently small disturbances of the initial value $a^*$ lead to half-paths that remain a small distance from $\mathcal{H}^*$. \ No newline at end of file