Chaos in 3D Brans-Dicke Model

The evidence of an accelerated expansion of the universe, observed by WMAP, opened the discussion of the general relativity eventual limits.
There are many options for alternative theories of gravity, and among them wecan cite scalar-tensor theories like supergravity, Kaluza-Klein theories, dual string theories, M-Theory, etc . One particular kind of scalar-tensor theory to describe an accelerate expansion of the universe, called Brans-Dicke theory, was proposed in the early sixties. This theory uses the principle of Mach and the hypothesis of Dirac, considering an eventual variation in time of the Newton's gravitational constant, thus ensuring the universality of free fall (equivalence principle). Most of the works which have been published in this theory up to now consider four flat dimensions, and some of them have tried to associate the scalar field of the Brans-Dicke theory as quintessence field as a type of K-essence field. Others have tried to find a solution for the observed accelerated expansion using a dimensional reduction of the 5D Brans-Dicke theory without matter. Concerning 3 dimensions, a broad study has been done in gravitational theories since the publication of BTZ Black Hole, motivated by the fact that 3D theories avoid some complications found in higher dimensions. However, there are not as many results about 3D scalar-tensor theories, and it would be interesting to find some results in this subject, more specifically in 3D Brans-Dicke theory. For instance, we can see some problems like the association of the scalar field of the Brans-Dicke theory to K-essence fields which models the dark energy. The goal in this paper is study the dynamical behavior of a 3D Brans-Dicke model. For this we use tools such as calculation of the exponent Lyaponov and analysis of phase space for the study of this dynamic system. From the results we identified regions whose theory has chaotic behavior.