  In the context of Degenerate Higher-Order Scalar-Tensor (DHOST) theories, we
study cosmological solutions and their stability properties. In particular, we
explicitly illustrate the crucial role of degeneracy by showing how the higher
order homogeneous equations in the physical frame (where matter is minimally
coupled) can be recast in a system of equations that do not involve higher
order derivatives. We study the fixed points of the dynamics, finding the
conditions for having a de Sitter attractor at late times. Then we consider the
coupling to matter field (described for convenience by a k-essence Lagrangian)
and find the conditions to avoid gradient and ghost instabilities at linear
order in cosmological perturbations, extending previous work. Finally, we apply
these results to a simple subclass of DHOST theories, showing that de Sitter
attractor conditions, no ghost and no gradient instabilities conditions (both
in the self-accelerating era and in the matter dominated era) can be
compatible.
