Dynamics of the universe with disformal coupling between the dark
  sectors
  We use the dynamical analysis to study the evolution of the universe at late
time for the model in which the interaction between dark energy and dark matter
is inspired by disformal transformation. We extend the analysis in the existing
literature by supposing that the disformal coefficient depends both on the
scalar field and its kinetic terms. We find that the dependence of the
disformal coefficient on the kinetic term of scalar field leads to two classes
of the scaling fixed points that can describe the acceleration of the universe
at late time. The first class exists only for the case where the disformal
coefficient depends on the kinetic terms. The fixed points in this class are
saddle points unless the slope of the conformal coefficient is sufficiently
large. The second class can be viewed as the generalization of the fixed points
studied in the literature. According to the stability analysis of these fixed
points, we find that the stable fixed point can take two different physically
relevant values for the same value of the parameters of the model. These
different values of the fixed points can be reached for different initial
conditions for the equation of state parameter of dark energy. We also discuss
the situations in which this feature disappears.
