  A simple second quantization model is used to describe a two-mode
Bose-Einstein condensate (BEC), which can be written in terms of the generators
of a SU(2) algebra with three parameters. We study the behaviour of the
entanglement entropy and localization of the system in the parameter space of
the model. The phase transitions in the parameter space are determined by means
of the coherent state formalism and the catastrophe theory, which besides let
us get the best variational state that reproduces the ground state energy. The
entanglement entropy is determined for two recently proposed partitions of the
two-mode BEC that are called separation by boxes and separation by modes of the
atoms. The entanglement entropy in the boxes partition is strongly correlated
to the properties of localization in phase space of the model, which is given
by the evaluation of the second moment of the Husimi function. To compare the
fitness of the trial wavefunction its overlap with the exact quantum solution
is evaluated. The entanglement entropy for both partitions, the overlap and
localization properties of the system get singular values along the separatrix
of the two-mode BEC, which indicates the phase transitions which remain in the
thermodynamical limit, in the parameter space.
