  In the gravitational effective theories including higher curvature terms,
cosmological solutions can have nontrivial de Sitter fixed points. We study
phenomenological implications of such points, focusing on a theory in which a
massive scalar field is nonminimally coupled to the Euler density. We first
analyze the phase portrait of the dynamical system and show that the fixed
point can be a sink or a saddle, depending on the strength of the coupling.
Then, we compute the perturbation spectra generated in the vicinity of the
fixed point in order to investigate whether the fixed point may be considered
as cosmic inflation. We find parameter regions that are consistent with the
cosmological data, given that the anisotropies in the cosmic microwave
background are seeded by the fluctuations generated near the fixed point.
Future observation may be used to further constrain the coupling function of
this model. We also comment briefly on the swampland conjecture.
