Dynamical system approach for the modified Brans–Dicke theory

A modified Brans–Dicke theory (abbreviated as GBD) is proposed by generalizing the Ricci scalar [Formula: see text] to an arbitrary function [Formula: see text] in the original BD action. It can be found that the GBD theory has some interesting properties, such as solving the problem of PPN value without introducing the so-called chameleon mechanism (comparing with the [Formula: see text] modified gravity), making the state parameter to crossover the phantom boundary: [Formula: see text] without introducing the negative kinetic term (comparing with the quintom model). In the GBD theory, the gravitational field equation and the cosmological evolutional equations have been derived. In the framework of cosmology, we apply the dynamical system approach to investigate the stability of the GBD model. A five-variable cosmological dynamical system and three critical points ([Formula: see text], [Formula: see text], [Formula: see text]) are obtained in the GBD model. After calculation, it is shown that the critical point [Formula: see text] corresponds to the radiation dominated universe and it is unstable. The critical point [Formula: see text] is unstable, which corresponds to the geometrical dark energy dominated universe. While for case of [Formula: see text], according to the center manifold theory, this critical point is stable, and it corresponds to geometrical dark energy dominated de Sitter universe ([Formula: see text]).