Homoclinic chaos in the dynamics of a general Bianchi IX model

TL;DR

This paper investigates chaotic dynamics in a Bianchi IX cosmological model with dust and a positive cosmological constant, revealing homoclinic chaos, fractal basin boundaries, and an asymptotic DeSitter attractor.

Contribution

It demonstrates the presence of homoclinic chaos and fractal basin boundaries in the phase space of a general Bianchi IX model, a novel insight into its complex dynamics.

Findings

Homoclinic transversal crossings indicate chaos.

Fractal basin boundaries characterize initial condition sensitivity.

An asymptotic DeSitter attractor influences long-term behavior.

Abstract

The dynamics of a general Bianchi IX model with three scale factors is examined. The matter content of the model is assumed to be comoving dust plus a positive cosmological constant. The model presents a critical point of saddle-center-center type in the finite region of phase space. This critical point engenders in the phase space dynamics the topology of stable and unstable four dimensional tubes $R \times S^3$, where $R$ is a saddle direction and $S^3$ is the manifold of unstable periodic orbits in the center-center sector. A general characteristic of the dynamical flow is an oscillatory mode about orbits of an invariant plane of the dynamics which contains the critical point and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of tubes (one stable, one unstable) emerging from the neighborhood of the critical point towards the FRW singularity have homoclinic transversal crossings. The homoclinic intersection manifold has topology $R \times S^2$ and is constituted of homoclinic orbits which are bi-asymptotic to the $S^3$ center-center manifold. This is an invariant signature of chaos in the model, and produces chaotic sets in phase space. The model also presents an asymptotic DeSitter attractor at infinity and initial conditions sets are shown to have fractal basin boundaries connected to the escape into the DeSitter configuration (escape into inflation), characterizing the critical point as a chaotic scatterer.