  We study a class of 2-dimensional Hamiltonian systems
$H(x,y,p_x,p_y)=\frac12(p_x^2+p_y^2) +V(x,y)$ in which the plane $x$=$p_x$=0 is
invariant under the Hamiltonian flow, so that straight-line librations along
the y axis exist, and we also consider perturbations $\delta H=\delta\cdot
F(x,y,p_x,p_y)$ that preserve these librations. We describe a procedure for the
analytical calculation of partial derivatives of the Poincar\'e map. These
partial derivatives can be used to predict the bifurcation behavior of the
libration, in particular to distinguish between transcritical and fork-like
bifurcations, as was mathematically investigated in [1] and numerically studied
in [2].
  [1] K. J\"anich, arXiv.org/abs/0710.3464 [2] M. Brack and K. Tanaka,
arXiv:0705.0753
