How to numerically evaluate index of fixed point?

Question

I want to take a brute force approach to determining whether a system of 2 ODEs can support oscillations/closed orbits with different combinations of parameters. After a little analysis, I know the system can only have 1 fixed point and, when it exists, I can reliably find it. Now I want to numerically compute its index, but can't figure out how to formulate the integral. I see from this question the following integral $$ \frac{1}{2\pi}\oint \frac{\dot{y}\ddot{x} - \dot{x}\ddot{y}}{\dot{x}^2 + \dot{y}^2}dt \, . $$ Where does this come from? How can I compute this integral for a little circle around a fixed point? Is there any existing code that, given $\dot{x}$ and $\dot{y}$, numerically evaluates such an integral?