Simple Dynamics on the Brane

TL;DR

This paper uses dynamical systems methods to analyze the evolution of Randall-Sundrum brane models, revealing stable, qualitatively similar behaviors to general relativity but with notable quantitative differences.

Contribution

It introduces a simplified 2D Hamiltonian dynamical system approach that avoids degeneracy of critical points, enhancing the analysis of Randall-Sundrum models.

Findings

Models exhibit qualitatively similar evolution to general relativity.

The dynamical system is structurally stable and free from degeneracy.

Quantitative differences distinguish brane models from classical gravity.

Abstract

We apply methods of dynamical systems to study the behaviour of the Randall-Sundrum models. We determine evolutionary paths for all possible initial conditions in a 2-dimensional phase space and we investigate the set of accelerated models. The simplicity of our formulation in comparison to some earlier studies is expressed in the following: our dynamical system is a 2-dimensional Hamiltonian system, and what is more advantageous, it is free from the degeneracy of critical points so that the system is structurally stable. The phase plane analysis of Randall-Sundrum models with isotropic Friedmann geometry clearly shows that qualitatively we deal with the same types of evolution as in general relativity, although quantitatively there are important differences.