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enter image description here

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 enter image description here

Unknown x
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    I think that there are very pathological examples of what you are looking for. However, one usually defines asymptotical stability as stability + the condition you wrote – Aitor Iribar Lopez Jan 07 '22 at 15:57
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    Look at this question, there you can find the example you want https://math.stackexchange.com/questions/1046071/closed-orbits-of-dynamical-systems – Aitor Iribar Lopez Jan 07 '22 at 15:58

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