It is well known that for a typical halo orbit around L1 or L2 libration point in circular restricted three-body problem its monodromy matrix has eigenvalues of the following form:

• lambda1 > 1
• lambda2 = 1 / lambda1 < 1
• lambda3 = lambda4 = 1
• lambda5 = lambda6*, |lambda5| = |lambda6| = 1

It is also well known that eigenvectors associated with lambda1 and lambda2 linearly approximate directions along the unstable and stable invariant manifolds, respectively. What about other lambdas?

As I understand, the compex pair (lambda5 and lambda6) is associated with a two-dimensional invariant subspace in which vectors rotate by the angle rho, where lambda5 = exp(i*rho). Am I right?

What about lambda3 and lambda4? Since the system of equations in CR3BP is Hamiltonian and autonomous, each periodical orbit has at least 2 eigenvalues equal to +1. So, in our case, the algebraic multiplicity is 2. What about geometric multiplicity? As I understand, there is at least one eigenvector, assotiated with 1, it is the direction along the orbit. Is it true that another independent eigenvector (if any) is directed along the family of halo orbits?