Spectral Stability of Periodic Solutions for Hamiltonian Systems with z₂ Symmetry

The condition for spectral stability of periodic solutions is established by carefully analyzing the finite determinacy of the Birkhoff normal form of a given symmetric Hamiltonian system. The Z₂ symmetry and the nondegeneracy of the 4-jet of the normal form play a key role in the proof. It is interesting to see how the coefficients of the truncated 4-jet of the Hamiltonian determine the existence and the spectral stability of normal modes (periodic solutions) of the full Hamiltonian system. There are many applications of the results. In particular, the stability of periodic and quasi-periodic motions of a solid body [1] will be analyzed in the sequel by using these results.