  We develop the method of adiabatic tracking for photo- and
magneto-association of Bose-Einstein atomic condensates with models that
include Kerr type nonlinearities. We show that the inclusion of these terms can
produce qualitatively important modifications in the adiabatic dynamics, like
the appearance of bifurcations, in which the trajectory that is being tracked
loses its stability. As a consequence the adiabatic theorem does not apply and
the adiabatic transfer can be strongly degraded. This degradation can be
compensated by using fields that are strong enough compared with the values of
the Kerr terms. The main result is that, despite these potentially detrimental
features, there is always a choice of the detuning that leads to an efficient
adiabatic tracking, even for relatively weak fields.

Quantum Physics;Quantum Gases