Is the critical point $C$ said to be stable and Asymptotically stable are same? I could find an example for a stable equilibrium point which is not asymptotically stable
$$x'=-y$$
$$y'=x$$
I can prove that $(X,Y)=(0,0)$ is stable. For an $\epsilon$, I could able to find $\delta=\epsilon$ such that when ever $||(x(0),y(0))-(0,0)||<\delta=\epsilon$. After solving, we get $x(t)^2+y(t)^2=K$. We know that when $(x(0),y(0))$ is the initial condition, Given solution will be $x(t)^2+y(t)^2=x(0)^2+y(0)^2.$ Which is less than $\epsilon$. But Doesn't satisfy the definition of asymptotically stable.
Conversely, I am not able to find an explicit example of a critical point $C$ said to be Asymptotically stable but not stable.
My attempt
