On a dynamical system linked to the BKL scenario

TL;DR

This paper analyzes a six-dimensional dynamical system related to the BKL cosmological singularity scenario, revealing integrability properties of its reduced forms and providing explicit solutions involving exponential and Painlevé functions.

Contribution

It demonstrates the integrability of reduced systems derived from a nonintegrable six-dimensional model and explicitly constructs their solutions.

Findings

Reduced systems are integrable despite the original system's nonintegrability.

Explicit solutions involve exponential functions and Painlevé functions.

The six-dimensional system models the BKL cosmological singularity scenario.

Abstract

We consider the six-dimensional dynamical system in three components introduced by Ryan to describe the scenario of Belinskii, Khalatnikov and Lifshitz to the cosmological singularity when the spatial metric tensor is not diagonal. Despite its nonintegrability, recently proven by Goldstein and Piechocki, the three four-dimensional systems defined by canceling one of the three components happen to be integrable. We express their general solution as a rational function of, respectively, two exponential functions, a third Painlev\'e function, two exponential functions.