International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 561-570
doi:10.1155/S0161171295000718
On the numerical solution of perturbed bifurcation problems
Department of Mathematics, University of Wisconsin - Eau Claire, Eau Claire 54702, WI, USA
Received 3 February 1993; Revised 22 March 1994
Copyright © 1995 M. B. M. Elgindi and R. W. Langer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.