Singular Perturbations of Bifurcations with Multiple Independent Bifurcation Parameters

Abstract

There are several nonlinear bifurcation problems which involve multiple bifurcation parameters (“specified inputs”). Often there is a corresponding “perturbed” problem associated with the bifurcation problem which models imperfections. Instead of bifurcation, the perturbed problem exhibits a smooth but rapid transition in the critical range of the bifurcation parameter. A new generalization of the single bifurcation parameter method of Matkowsky and Reiss, SPB (“Singular Perturbations of Bifurcations”) [SIAM J. Appl. Math., 33 (1977), pp. 230–255] is employed to solve the problem of the slightly crooked rotating elastica (shaft) subject to a dead load applied at the ends.