Bifurcations of the Equilibrium of a Spherical Double Pendulum at a Multiple Eigenvalue

Abstract

The loss of stability of the downhanging equilibrium position of tubes conveying fluid and allowing for a three-dimensional motion has been studied in [1–3], where both continuous and discrete models for the tube have been used. This problem is particularly interesting from the standpoint of stability theory because due to certain symmetries in the system which we will explain below the critical eigenvalues (i.e. eigenvalues with zero real part; see chapter 3) have double multiplicity. As we want to analyse the nonlinear problem in its post-bifurcational behavior we shall perform a reduction of the n-dimensional problem to a system of bifurcation equations on the center manifold, the dimension of which is equal to the number of critical eigenvalues. Hence at the first glance, due to the doubled dimension, a very complicated situation is found for the derivation of the bifurcation equations. However, due to the symmetry properties, which also the bifurcation equations on the center manifold must fulfill, they are pretty simple, such that these two effects, namely the higher multiplicity of the critical eigenvalue and the symmetry properties of the bifurcation equations compensate each other in some sense.