  We investigate the cosmological dynamics of interacting dark energy models in
which the interaction function is a nonlinear in terms of the energy densities.
Considering explicitly the interaction between a pressureless dark matter and a
scalar field, minimally coupled to Einstein gravity, we explore the dynamics of
the spatially flat FLRW universe for the exponential potential of the scalar
field.. We perform the stability analysis for the three nonlinear interaction
models of our consideration through the analysis of critical points and we
investigate the cosmological parameters and we discuss the physical behaviour
at the critical points. From the analysis of the critical points we find a
number of possibilities that include the stable late time accelerated solution,
$w$CDM-like solution, radiation-like solution and moreover the unstable
inflationary solution as well.
