Show that the equilibrium point $\displaystyle x_0=0$ for the differential equation $\displaystyle x^{\prime}=0$ is stable but not asymptotically stable.

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- Sep 8th 2010, 09:51 AMjoestevensStable by but not Asymptotically Stable
Show that the equilibrium point $\displaystyle x_0=0$ for the differential equation $\displaystyle x^{\prime}=0$ is stable but not asymptotically stable.

- Sep 8th 2010, 05:29 PMJJMC89
- Sep 8th 2010, 05:30 PMjoestevens
- Sep 9th 2010, 04:26 AMJester
In this problem you can actually integrate and find the solution

$\displaystyle x(t) = x_0$

Now, with $\displaystyle x_0 = 0$ gives your solution. Now perturb it so

$\displaystyle x^{*}(t) = x_1$.

So,

$\displaystyle | x(t) - x^{*}(t)| = x_1 $ for all t (this a stable)

However, we don't have for some $\displaystyle \delta > 0$

$\displaystyle | x(t) - x^{*}(t)| \le \delta \; \text{for}\; t > t*$ (this a asymptotic stability).