We perform an analytical study of the bifurcation of the halo orbits around
the collinear points L1​, L2​, L3​ for the circular, spatial, restricted
three--body problem. Following a standard procedure, we reduce to the center
manifold constructing a normal form adapted to the synchronous resonance.
Introducing a detuning, which measures the displacement from the resonance and
expanding the energy in series of the detuning, we are able to evaluate the
energy level at which the bifurcation takes place for arbitrary values of the
mass ratio. In most cases, the analytical results thus obtained are in very
good agreement with the numerical expectations, providing the bifurcation
threshold with good accuracy. Care must be taken when dealing with L3​ for
small values of the mass-ratio between the primaries; in that case, the model
of the system is a singular perturbation problem and the normal form method is
not particularly suited to evaluate the bifurcation threshold.Comment: 35 pages, 3 figures, updated version accepted for publication on
Physica