Thread: Stable by but not Asymptotically Stable

Show that the equilibrium point for the differential equation is stable but not asymptotically stable.

Originally Posted by joestevens

Show that the equilibrium point for the differential equation is stable but not asymptotically stable.

I don't know exactly how to do this but you probably need to use the ε, δ definition of stable and asymptotically stable.

Originally Posted by JJMC89

I don't know exactly how to do this but you probably need to use the ε, δ definition of stable and asymptotically stable.

I got as far as knowing that but I don't know where to do. Anyone have any suggestions as to how to do this?

In this problem you can actually integrate and find the solution



Now, with gives your solution. Now perturb it so

.

So,

for all t (this a stable)

However, we don't have for some

(this a asymptotic stability).