Bianchi IX model: Reducing phase space

TL;DR

This paper analyzes the complex phase space structure of the nondiagonal Bianchi IX cosmological model near singularities, employing dynamical systems and canonical transformations to reduce dimensionality and better understand its dynamics.

Contribution

It introduces a method to reduce the phase space of the Bianchi IX model using canonical transformations and analyzes the nature of critical points at infinity.

Findings

Critical points are nonhyperbolic and appear at infinities.

Reduction of the symplectic form simplifies the dynamics.

Method shows promise for further dynamical analysis.

Abstract

The mathematical structure of higher-dimensional physical phase spaces of the nondiagonal Bianchi IX model is analyzed in the neighborhood of the cosmological singularity by using dynamical system methods. Critical points of the Hamiltonian equations appear at infinities and are of a nonhyperbolic type, which is a generic feature of the considered singular dynamics. The reduction of the kinematical symplectic 2-form to the constraint surface enables the determination of the physical Hamiltonian. This procedure lowers the dimensionality of the dynamics arena. The presented analysis of the phase space is based on canonical transformations. We test our method for the specific subspace of the physical phase space. The obtained results encourage further examination of the dynamics within our approach.