  In the last decade the Sznajd Model has been successfully employed in
modeling some properties and scale features of both proportional and majority
elections. We propose a new version of the Sznajd model with a generalized
bounded confidence rule - a rule that limits the convincing capability of
agents and that is essential to allow coexistence of opinions in the stationary
state. With an appropriate choice of parameters it can be reduced to previous
models. We solved this new model both in a mean-field approach (for an
arbitrary number of opinions) and numerically in a Barabasi-Albert network (for
three and four opinions), studying the transient and the possible stationary
states. We built the phase portrait for the special cases of three and four
opinions, defining the attractors and their basins of attraction. Through this
analysis, we were able to understand and explain discrepancies between
mean-field and simulation results obtained in previous works for the usual
Sznajd Model with bounded confidence and three opinions. Both the dynamical
system approach and our generalized bounded confidence rule are quite general
and we think it can be useful to the understanding of other similar models.
