Problem 6.16 Ordinary Differential Equations and Dynamical Systems Teschl While reading the book Ordinary Differential Equations and Dynamical Systems by Gerald Teschl, I got stuck at Problem 6.16. It states:
Consider the system
\begin{eqnarray*}
\dot{x} = x-y-x(x^2+y^2)+\dfrac{xy}{\sqrt{x^2+y^2}} \\
\dot{y} = x+y-y(x^2+y^2)-\dfrac{x^2}{\sqrt{x^2+y^2}}.
\end{eqnarray*}
Show that $(1,0)$ is not stable even though $\lim\limits_{t \to +\infty} |\phi(t,x) - x_0| = 0$.
I was able to show the second part, but i could not show that $(1,0)$ is not stable. By plotting the system on mathlab I could see that it isn't stable, but how do I show it using the definition of stability?

Hint.
From
\begin{eqnarray*}
\dot{x} = x-y-x(x^2+y^2)+\dfrac{xy}{\sqrt{x^2+y^2}} \\
\dot{y} = x+y-y(x^2+y^2)-\dfrac{x^2}{\sqrt{x^2+y^2}}.
\end{eqnarray*}
we have
\begin{eqnarray*}
x\dot{x} = x^2-xy-x^2(x^2+y^2)+\dfrac{x^2y}{\sqrt{x^2+y^2}} \\
y\dot{y} = xy+y^2-y^2(x^2+y^2)-\dfrac{x^2y}{\sqrt{x^2+y^2}}.
\end{eqnarray*}
and after addition
$$
(x^2+y^2)'=(x^2+y^2)(1-(x^2+y^2))
$$
or making $r=\sqrt{x^2+y^2}$
$$
\dot r = r(1-r^2)
$$
