# Generalized Brans-Dicke Theory: A Dynamical Systems Analysis

## TL;DR

This paper analyzes the stability of generalized Brans-Dicke cosmology with a variable coupling parameter, revealing conditions under which solutions resemble general relativity and exploring effects of scalar potentials.

## Contribution

It extends stability analysis to a generalized Brans-Dicke theory with a variable coupling parameter and examines the impact of scalar potentials on stability conditions.

## Key findings

- Most solutions approach general relativity in the infinite $mbda$ limit.
- Stability depends on the matter distribution and the form of the scalar potential.
- Radiation distribution exhibits unique stability behavior compared to other fluids.

## Abstract

The stability criteria for the generalized Brans-Dicke cosmology in a spatially flat, homogeneous and isotropic cosmological model is discussed in the presence of a perfect fluid. The generalization comes through the channel that the Brans-Dicke coupling parameter $\omega$ is allowed to be a function of the scalar field $\phi$. This generalization can lead to a host of scalar-tensor theories of gravity for various choices of $\omega = \omega (\phi)$. A very interesting general result has been found. Excepting for the case of a radiation distribution as the choice of the fluid, all other solutions find a natural habitat in the corresponding solutions in general relativity in an infinite $\omega$ limit. For the radiation distribution, the dependence of stability on $\omega$ is a bit obscure. If a scalar potential, function of the Brans-Dicke scalar field, is added to the action, the requirement of an infinite $\omega$ for stability is relaxed for a matter distribution with a non-zero trace whereas it becomes a possibility for a radiation distribution.