Dynamical properties of cosmological models in Horndeski theory

In this paper, we study the dynamics of homogeneous isotropic cosmological models with a spatially flat metric in theories of gravity with a scalar field that is non-minimally coupled to curvature. The non-minimal coupling here is characterized by the presence in the action functional terms of the form ξRϕ2 and ηGμν∇μϕ∇νϕ (kinetic coupling). We will consider a theory that includes the quadratic potential of a scalar field. Due to the non-linearity of the resulting dynamic equations, in our analysis we will primarily be interested in asymptotic behavior, and also use numerical integration, including to represent the dynamics of the model in the form of a phase portrait.