# Switch branch in bifurcation

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 $a$?

• A basis for the nullspace of the Jacobian at the point of bifurcation should give you a set of directions you can use to explore families of solutions via something like a continuation method. – Geoff Oxberry Sep 29 '13 at 2:57
• So, since rank(N(J))=1 in my case, I got a basis vector u for the nullspace. If my solution is (x0,a0) at the bifurcation how do I generate a new initial guess for my system that will converge to a solution on the new branch? Thanks for the answer btw. – chemeng Sep 29 '13 at 3:07
• If $a_{0}$ is where your bifurcation occurs, and $v$ is the normalized vector in the nullspace of the Jacobian matrix, then I'd try guesses of the form $F(x_{0} + cv, a_{0})$, where $c$ is some small constant. You'll probably want to try both positive and negative values of $c$, and you may need to perturb $a$ slightly as well. Books on continuation methods will provide more detailed advice. – Geoff Oxberry Sep 30 '13 at 17:42