  We present a rigorous analysis of the role and uses of the adiabatic
invariant in the Mixmaster dynamical system. We propose a new invariant for the
global dynamics which in some respects has an improved behaviour over the
commonly used one. We illustrate its behaviour in a number of numerical
results. We also present a new formulation of the dynamics via Catastrophe
Theory. We find that the change from one era to the next corresponds to a fold
catastrophe, during the Kasner shifts the potential is an Implicit Function
Form whereas, as the anisotropy dissipates, the Mixmaster potential must become
a Morse 0--saddle. We compare and contrast our results to many known works on
the Mixmaster problem and indicate how extensions could be achieved. Further
exploitation of this formulation may lead to a clearer understanding of the
global Mixmaster dynamics.
