Generic bifurcations of pendula

Dellnitz, M., J. E. Marsden, I. Melbourne and J. Scheurle

Int. Series on Num. Math., Birkhäuser, (G. Allgower and K. Böhmer and M. Golubitsky, eds.), 104, (1992), 111-122

Abstract:

In a parameter dependent Hamiltonian system, an equilibrium might lose its stability via a socalled Hamiltonian Krein-Hopf bifurcation: Two pairs of purely imaginary eigenvalues of the linearized system collide (1-1 resonance) and split off the imaginary axis into the complex plane. In the following we will refer to this scenario as the splitting case. It is well known that in one parameter problems without external symmetry this is the only eigenvalue behavior that generically occurs in 1-1 resonances.
When there is symmetry present, the situation changes. Under certain circumstances the eigenvalues might also pass while remaining on the imaginary axis. In this case the linear stability properties of the corresponding equilibrium do not change and in this sense the collision is not "dangerous" as in the splitting case.
The question arises naturally whether these essentially different eigenvalue movements can be characterized so that the occurrence of the one or the other in a given system could in principle be predicted. The answer to this question is given in [3] by Dellnitz, Melbourne and Marsden. There the generic movement of eigenvalues through a 1-1 resonance is completely classified by use of group theory and energetics.
The main purpose of this paper is to show the usefulness of this type of result for analysing the dynamical behavior of mechanical systems. We describe briefly the main result of [3] in Section 2 and consider rotating pendular problems in Section 3.
These examples clearly point out the fact that in specific mechanical systems, both passing and splitting can occur generically. In [3] this behavior is explained in the context of systems with symplectic symmetries. The examples suggest there is a corresponding result for systems with antisymplectic symmetries as well.