In this work, our target is to analyze the dynamics around the $1:-1$
resonance which appears when a family of periodic orbits of a real
analytic three-degree of freedom Hamiltonian system changes its
stability from elliptic to a complex hyperbolic saddle passing
through degenerate elliptic. Our analytical approach consists of
computing, in a constructive way and up to some given
arbitrary order, the normal form around that resonant (or
\emph{critical}) periodic orbit.
Hence, dealing with the normal form itself and the differential
equations related to it, we derive the generic existence of a
two-parameter family of invariant 2D tori which bifurcate from the
critical periodic orbit. Moreover, the coefficient of the normal form
that determines the stability of the bifurcated tori is
identified. This allows us to show the Hopf-like character of the
unfolding: elliptic tori unfold ``around'' hyperbolic periodic orbits
(case of \emph{direct} bifurcation) while normal hyperbolic tori
appear ``around'' elliptic periodic orbits (case of \emph{inverse}
bifurcation). Further, a global description of the dynamics of the
normal form is also given.
