Multiple Eigenvalues Lead to Secondary Bifurcation

Abstract

Bifurcation problems are considered where the primary bifurcation points are functions of a parameter $\tau $. It is shown that a multiple bifurcation point, which occurs for $\tau = \tau _0 $, may “split” into two (or more) simple primary bifurcation points and several secondary bifurcation points as $\tau $ varies from $\tau _0 $. These secondary points move along one or more of the primary branches as $\tau $ varies. This is shown to occur for a simple two-degrees-of-freedom system, which is a model for plate and rod buckling. This analysis shows that the splitting occurs in several different and physically significant ways. A new perturbation method is presented for analyzing problems which cannot be explicitly solved. The method is applied to study secondary bifurcations in the axisymmetric buckling of spherical shells. The paper concludes with a brief discussion of some physical problems where this new phenomenon may occur and where the perturbation method may be applicable.