  We discuss dynamics of approximate adiabatic invariants in several nonlinear
models being related to physics of Bose-Einstein condensates (BEC). We show
that nonadiabatic dynamics in Feshbach resonance passage, nonlinear
Landau-Zener (NLZ) tunnelling, and BEC tunnelling oscillations in a double-well
can be considered within a unifying approach based on the theory of separatrix
crossings. The separatrix crossing theory was applied previously to some
problems of classical mechanics, plasma physics and hydrodynamics, but has not
been used in the rapidly growing BEC-related field yet. We derive explicit
formulas for the change in the action in several models. Extensive numerical
calculations support the theory and demonstrate its universal character. We
also discovered a qualitatively new nonlinear phenomenon in a NLZ model which
we propose to call {\em separated adiabatic tunnelling}
