  We consider a cosmological model consisting of two scalar fields defined in
the hyperbolic plane known as hyperbolic inflation. For the background space,
we consider a homogeneous and isotropic spacetime with nonzero curvature. We
study the asymptotic behaviour of solutions and we search for attractors in the
expanding regime. We prove that two hyperbolic inflationary stages are stable
solutions that can solve the flatness problem and describe acceleration for
both open and closed models, and additionally we obtain a Milne-like attractor
solution for the open model. We also investigate the contracting branch
obtaining mirror solutions with the opposite dynamical behaviours.
