I am working on a problem from Differential Equations and Dynamical Systems and I can not get past a simple eigenvalue problem. I need to show the origin is an unstable focus for the following system: $$ \begin{align} \dot{x} &= -y+x(r^{4}-3r^{2}+1)\\ \dot{y} &= x+y(r^{4}-3r^{2}+1) \end{align} $$
Now using $r^{2} = x^{2}+y^{2}$ and $r\dot{r} = x\dot{x} + y \dot{y}$, I have already established that the origin is an equilibrium point by setting $\dot{x}=\dot{y}=0$ and finding this holds when $\frac{y}{x} = -\frac{x}{y}$. Also we find $$ \dot{r} = r^{5}-3r^{3}+r.$$
Now I need to show this is an unstable focus.
In the book they simply state the eigenvalues for the linear part of the system, $Df(0,0)$, are $\pm 2i$ but I can not seem to find this. What form does $Df(0,0)$ have? Do we use $f(\dot{x},\dot{y})$, $f(x,y)$ or $f(r, \theta)$? For both parametrizations I can not find the stated eigenvalues.
Any help is much appreciated.
The linear approximation is
$$ \cases{\dot x = x-y\\ \dot y = x + y} $$
and the jacobian $D f(0,0) = \left( \begin{array}{cc} 1 & -1 \\ 1 & 1 \\ \end{array} \right)$ has eigenvalues $(1+i,1-i)$ characterizing an unstable spiral source at $(0,0)$
NOTE
From
$$ \dot r = r(r^4-3r^2+1)=r\left((r^2-1)^2-r^2 \right) $$
with $r \ge 0$ we have a $\dot r$ behavior shown in the following plot
such that for $0< r < \frac{1}{2} \left(\sqrt{5}-1\right)$ we have $\dot r > 0$ and for $\frac{1}{2} \left(\sqrt{5}-1\right) < r < \frac{1}{2} \left(\sqrt{5}+1\right)$ we have $\dot r \lt 0$ and finally for $r > \frac{1}{2} \left(\sqrt{5}+1\right)$ we have $\dot r > 0$ so at the origing we have an unstable source. The symmetry and the change of sign for $\dot r$ leave us to suspect the existence of limit cycles as reflected into the following stream plot.