I am trying to implement an algorithm that finds (possibly) all solutions of a system of nonlinear polynomial equations $$F(X) = 0$$ I thought about using the (convex) homotopy $$H(X,T) = TF(X) + (1-T)G(X_0)$$ with $X_0$ a known solution to $G(X_0)=0$. By increasing $T$ iteratively from $0$ to $1$, the sought solution of $F(X)$ will be found.
During the hike on that path, I expect to encounter points of different bifurcation types. I hoped to use some of the many well-implemented numerical continuation algorithms that are available on the internet and would allow me to detect those points with their belonging paths.
I'm currently stuck at getting the numerical continuation running on the homotopy. To be more precise, the paths of solutions during numerical continuation is far from being a solution to $H(X,1)=0$. I do have a solid background in programming but only mediocre knowledge of mathematics. Therefore I hoped someone could review my approach and answer the following questions:
Is numerical continuation applicable to the homotopy method? I see that there are some differences compared to the continuation of an ODE (i.e. in the predictor/corrector scheme) but isn't the approach the same?
If not, can one detect the above-mentioned bifurcation points differently?