Problem 6.16 Ordinary Differential Equations and Dynamical Systems Teschl

Question

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?

Answer 1

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 $r=\sqrt{x^2+y^2}$

$$ \dot r = r(1-r^2) $$