On the numerical solution of perturbed bifurcation problems

Some numerical schemes, based upon Newton's and chord methods, for the computations of the perturbed bifurcation points as well as the solution curves through them, are presented. The initial guesses for Newton's and chord methods are obtained using the local analysis techniques and proved to fall into the neighborhoods of contraction for these methods. In applications the perturbation parameter represents a physical quantity and it is desirable to use it to parameterize the solution curves near the perturbed bifurcation point. In this regard, it is shown that, for certain classes of the perturbed bifurcation problems, Newton's and chord methods can be used to follow the solution curves in a neighborhood of the perturbed bifurcation point while the perturbation parameter is kept fixed.