Adiabatic invariants and Mixmaster catastrophes

Abstract

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.