I have a system of nonlinear equations $F(x,a) = 0$ and I know that at a specific point $a_c$ a bifurcation occurs, thus the Jacobian becomes singular. How can I switch branches and start following a new solution path as I increment $a$?
When the Jacobian matrix becomes singular and there is no simple fold, so the continuation point is not regular, multiple branches emanate from such a point. In that case, to switch to a different branch than the one that follows the direction vi (as defined above) the continuation must be perturbed. The algebraic bifurcation equation gives the necessary direction. One can often also perturb the system and then try to converge parallel to the original branch; typically the second branch is then found.
There are also some packages mentioned on the site which implement automatic branch switching (e.g., AUTO).