From b7ee0d14598a4d335dd65ef979b600dfc9e306a9 Mon Sep 17 00:00:00 2001 From: Youwen Wu Date: Thu, 6 Mar 2025 02:09:29 -0800 Subject: [PATCH] auto-update(nvim): 2025-03-06 02:09:29 --- .../by-course/math-4b/course-notes/main.typ | 16 ++++++++++++++++ 1 file changed, 16 insertions(+) diff --git a/documents/by-course/math-4b/course-notes/main.typ b/documents/by-course/math-4b/course-notes/main.typ index 7a14db4..b9af230 100644 --- a/documents/by-course/math-4b/course-notes/main.typ +++ b/documents/by-course/math-4b/course-notes/main.typ @@ -1330,3 +1330,19 @@ We assume $ arrow(x)_p = vec(A e^t, B e^t) $ + += Phase plane, autonomous systems, stability + +== Classification of equilibria for $n=2$. + +Consider possible equilibria at $0$ for the system $arrow(x)' = A arrow(x)$ when $n = 2$. + +For real eigenvalues $r_1, r_2 != 0$: +- $r_1, r_2 < 0$ is an asymptotically stable node +- $r_1, r_2 > 0$ is an unstable node +- $r_1, r_2 < 0$ is an unstable saddle + +For complex eigenvalues $lambda plus.minus i mu, mu != 0$ +- $lambda = 0$ is a center, stable +- $lambda < 0$ is asymptotically stable, spiral sink +- $lambda > 0$ is unstable, spiral source