Global stability analysis for cosmological models with nonminimally coupled scalar fields

We explore dynamics of cosmological models with a nonminimally coupled scalar field evolving on a spatially flat Friedmann-Lemaitre-Robertson-Walker background. We consider cosmological models including the Hilbert-Einstein curvature term and the N degree monomial of the scalar field nonminimally coupled to gravity. The potential of the scalar field is the n degree monomial or polynomial. We describe several qualitatively different types of dynamics depending on values of power indices N and n. We identify that three main possible pictures correspond to n < N, N < n < 2N and n > 2N cases. Some special features connected with the important cases of N = n (including the quadratic potential with quadratic coupling) and n = 2N (which shares its asymptotics with the potential of the Higgs-driven inflation) are described separately. A global qualitative analysis allows us to cover the most interesting cases of small N and n by a limiting number of phase-space diagrams. The influence of the cosmological constant to the global features of dynamics is also studied.