From c4985f97365c70b4c5b3f709ce6b1207fa4cd85e Mon Sep 17 00:00:00 2001
From: Longye Tian <longye.tian@outlook.com>
Date: Mon, 22 Jul 2024 12:34:43 +1000
Subject: [PATCH] add example of local stability

---
 lectures/scalar_dynam.md | 9 +++++++--
 1 file changed, 7 insertions(+), 2 deletions(-)

diff --git a/lectures/scalar_dynam.md b/lectures/scalar_dynam.md
index ab05d2a9..18c8568c 100644
--- a/lectures/scalar_dynam.md
+++ b/lectures/scalar_dynam.md
@@ -293,12 +293,17 @@ $$
 
 Obviously every globally stable steady state is also locally stable.
 
-We will see examples below where the converse is not true.
-
+Here is an example where the converse is not true.
 
+```{prf:example}
+Consider the self-map $g$ on $\mathbb{R}$ defined by $g(x)=x^2$. The fixed point $1$ is not stable.
 
+For example, $g^t (x)\to\infty$ for any $x>1$.
 
+However, $0$ is locally stable, because $-1<x<1$ implies that $g^t (x)\to 0$ as $t\to\infty$.
 
+Since we have more than one fixed point, $0$ is not globally stable.
+```
 
 
 ## Graphical analysis