Stability of nonlinear normal modes of symmetric Hamiltonian systems

Abstract

A nonlinear normal mode of a Hamiltonian system is a periodic solution near equilibrium with period close to that of a periodic trajectory of the linearised vector field. This paper is a sequel to the preceding paper in this issue, in which the authors developed methods for proving the existence of nonlinear normal modes of a Hamiltonian system that is invariant under the action of a compact Lie group. Here they show how to compute the spectral stability of these nonlinear normal modes. The methods are a natural extension of those used in the previous paper, namely, Birkhoff normal form and singularity theory. In particular they prove that under certain nondegeneracy conditions, Floquet exponents computed from a truncated Hamiltonian in normal form determine the spectral stability for the corresponding solutions of the full Hamiltonian. Applications include systems with O(2) and D_n symmetry, systems with five degrees of freedom and O(3) symmetry (modelling some vibrational modes of a liquid drop), and resonant time-reversible systems with Z₂ symmetry (such as the planar spring pendulum). They give explicit reductions to Birkhoff normal form for Hamiltonians of the form 'kinetic+potential' where the potential is invariant under O(2) or D_n. The analysis of systems with O(3) symmetry uses the dynamical invariance of fixed-point spaces of subgroups to simplify the computations considerably.