Show that the trajectory of the system move on a circle

Question

Consider the dynamical system given by $$\dot{x_1}=x_1^2-x_2^2, \qquad \dot{x_2}=2x_1x_2.$$ Show that the trajectories of the above system move along circles. Compute radius and center.

I have tried a few things, but no luck so far. There should be no need to compute trajectories.

Answer 1

The generic circle reads $(x_1-a)^2+(x_2-b)^2=r^2$ and the tangent space is

$$ (x_1-a)\dot x_1 + (x_2-b)\dot x_2 = 0 $$

so

$$ (x_1-a)(x_1^2-x_2^2)+2(x_2-b)x_1x_2=0 $$

or

$$ x_1^3+x_1x_2^2-ax_1^2+ax_2^2-2bx_1x_2=0 $$

now if $a=0$ we have

$$ x_1(x_1^2+x_2^2-2bx_2)=0 $$

Attached a stream plot with two circular orbits $x_1^2+x_2^2-2bx_2=0$ for $b=1$ (red) and for $b=-1$ (blue).