On the existence of regular orbit cylinders

Question

Let $(M,\omega,H)$ be a Hamiltonian system and assume that $\gamma$ is a periodic orbit on a regular energy hypersurface. Then the regular orbit cylinder theorem (see for example Abraham/Marsden: Foundations of Mechanics Theorem 8.2.2 or the book by Hofer/Zehnder on Hamiltonian Dynamics Proposition 2 on page 110) states the following:

If the flow of the Hamiltonian vector field $X_H$ at $\gamma(0)$ has precisely two Floquet multipliers equal to $1$, then $\gamma$ is contained in a whole one-parameter family of periodic orbits, i.e. an orbit cylinder.

Now there is a delicate Analysis of different bifurcation scenarios and various stability properties (see Abraham/Marsden chapter 8). My question: How "generic" is the condition of having precisely two Floquet multipliers equal to $1$? Equivalently, how likely is it that a Poincaré section map has precisely one eigenvalue equal to $1$?

Answer 1

The two multipliers correspond to the exponential growth rate of perturbations 1). Along the periodic orbit, and 2). Normal to energy hypersurface.

Unless there is additional symmetry in the system, two multipliers = 1 is the generic situation. A common example of symmetry is integral of motion of Hamiltonian system (of which the energy is an example). For each additional constant to motion, you should expect one more unity multiplier.

See more details in : Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D, 2003