auto-update(nvim): 2025-03-06 02:09:29

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