Adiabatic invariants and mixmaster catastrophes

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 behavior over the commonly used one. We illustrate its behavior 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.