I have the following system:
$\dot r=r(a-r^2)$
$\dot \theta=1$
I need to check the stability of the equilibrium point (0,0) for different values of a in $R$.
MY ATTEMPT:
If $a\lt r(0)^2$, then $\dot r \lt 0$, so, stable spiral.
If $a\gt r(0)^2$, then $\dot r \gt 0$, so, unstable spiral.
If $a= r(0)^2$,, then, r=r(0), so, centre., where $r(0)=r at t=0
Is my attempt correct? Please rectify if there is any mistake and give an explanation for the correction. Thankyou.