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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 $a$?

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  • $\begingroup$ 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. $\endgroup$ – Geoff Oxberry Sep 29 '13 at 2:57
  • $\begingroup$ 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. $\endgroup$ – chemeng Sep 29 '13 at 3:07
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    $\begingroup$ 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. $\endgroup$ – Geoff Oxberry Sep 30 '13 at 17:42
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From http://www.scholarpedia.org/article/Numerical_continuation#Branch_switching:

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).

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