Cosmological global dynamical systems analysis

TL;DR

This paper develops a comprehensive global dynamical systems analysis for cosmological models with scalar fields and matter, using advanced mathematical tools to understand their entire evolution.

Contribution

It introduces a center manifold analysis and constructs monotonic and Dulac functions to fully characterize the global dynamics, extending previous local studies.

Findings

Complete global phase space description achieved

Identification of bifurcation points and their effects

Proofs of asymptotic behaviors of cosmological solutions

Abstract

We consider a dynamical systems formulation for models with an exponential scalar field and matter with a linear equation of state in a spatially flat and isotropic spacetime. In contrast to earlier work, which only considered linear hyperbolic fixed point analysis, we do a center manifold analysis of the non-hyperbolic fixed points associated with bifurcations. More importantly though, we construct monotonic functions and a Dulac function. Together with the complete local fixed point analysis this leads to proofs that describe the entire global dynamics of these models, thereby complementing previous local results in the literature.