asymptotical stability

Question

If my system has the equilibirum point at $(x_1,0,0)$ where $x_1\in \mathbb{R}$. Why from this information I can say that the system is not asymptotically stable?

Answer 1

Assume it were. Then e.g. the equilibrium point $(0, 0, 0)$ is asymptotically stable. So there exists some $\varepsilon > 0$ such that $\displaystyle \lim_{t \rightarrow \infty} \varphi(t) = (0, 0, 0)$ for all solutions $\varphi$ with $\varphi(0) \in B_\varepsilon((0, 0, 0))$.

Now choose $x \in \mathbb{R} \setminus \lbrace 0 \rbrace$ such that $(x, 0, 0) \in B_\varepsilon((0, 0, 0))$. But then there exists $\delta > 0$ such that all solutions $\varphi$ with $\varphi(0) \in B_\delta((x, 0, 0))$ converge to $(x, 0, 0)$ as $t \rightarrow \infty$. But by construction, $ U:= B_\delta((x, 0, 0)) \cap B_\varepsilon((0, 0, 0))$ is nonempty. So solutions $\varphi(0) \in U$ converge to $(x, 0, 0)$ and $(0, 0, 0)$ as $t \rightarrow \infty$. But this makes no sense at all.