1-r), \\ \dot\theta &= \sin^2
      \left( \frac{\theta}{2} \right).\end{align*}  This system has two equilibrium points, one at
      $r^*=0,\theta^*=0$, and the other at $r^* = 1,\theta^*=0$.  The fixed
      point at zero is clearly unstable.  The fixed point with $r^*=1$ attracts
      all other trajectories, but it is not stable by any of our
      definitions.</p>

      <p>Although the equations are simpler in polar coordinates, it's useful to draw the vector field in Cartesian coordinates:</p>

    