Adiabatic invariants and Mixmaster catastrophes

TL;DR

This paper rigorously analyzes adiabatic invariants in the Mixmaster system, introduces a new invariant with improved behavior, and reformulates the dynamics using Catastrophe Theory to better understand the global behavior.

Contribution

It proposes a new global invariant and a novel formulation of Mixmaster dynamics via Catastrophe Theory, enhancing understanding of its complex behavior.

Findings

New invariant shows improved behavior over traditional ones

Mixmaster transitions correspond to fold catastrophes

Potential evolves from Implicit Function to Morse saddle

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.