Global stability analysis for cosmological models with non-minimally
  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.
