  We study cosmic evolution based on the fixed points in the dynamical analysis
of the Degenerate Higher-Order Scalar-Tensor (DHOST) theories. We consider the
DHOST theory in which the propagation speed of gravitational waves is equal to
the speed of light, the tensor perturbations do not decay to dark energy
perturbations, and the scaling solutions exist. The scaling fixed point
associated with late time acceleration of universe can be either stable or
saddle depending on the parameters of the theory. For some ranges of the
parameters, this scaling fixed point and field dominated fixed point can be
simultaneously stable. Cosmic evolution will reach either the scaling attractor
or the field dominated attractor depending on signs of time derivative of the
scalar field in the theory during the matter domination. The density parameter
of dark matter can be larger than unity before reaching the scaling attractor
if the deviation from the Einstein theory of gravity is too large. For this
DHOST theory, stabilities of $\phi$-matter-dominated epoch ($\phi$MDE) and
field dominated solutions are similar to the coupled dark energy models in
Einstein gravity even though gravity is described by different theories. In our
consideration, the universe can only evolve from the $\phi$MDE regime to the
field dominated regime. The ghost and gradient instabilities up to linear order
in cosmological perturbations have been investigated. There is no gradient
instability, while the ghost instability can be avoided for some range of the
parameters of the model.
