The 18-cycle in Bianchi $VI_{-1/9}^{^{*}}$: Combined Linear Local Passage and Numerical Simulation

Abstract

In this paper, we find an example for a periodic heteroclinic chain in Bianchi $VI_{-1/9}^{^{*}}$ that allows Takens Linearization at all base points. It turns out to be a "18-cycle'', i.e. involving a heteroclinic chain of 18 different base points at the Kasner circle. We then show that the combined cinear local passage at the 18-cycle is a contraction. This qualifies the 18-cycle as a candidate for proving the first rigorous convergence theorem in Bianchi $VI_{-1/9}^{^{*}}$. For the numerical simulation of solutions that follow such heteroclinic cycles, we use a variable-step, variable-order (VSVO) Adams-Bashforth-Moulton PECE solver in Matlab. We conclude with a discussion on how to proceed further in studying Bianchi cosmologies, and also discuss directions for future research in inhomogeneous (PDE-) cosmological models. This puts our results in a broader perspective. The appendix contains symbolic and computations done by Mathematica for examples discussed throughout the text.