  In this paper we study canonical scalar field models with a varying second
slow-roll parameter, that allow transitions between constant-roll eras. In the
models with two constant-roll eras it is possible to avoid fine-tunings in the
initial conditions of the scalar field. We mainly focus on the stability of the
resulting solutions and we also investigate if these solutions are attractors
of the cosmological system. We shall calculate the resulting scalar potential
and by using a numerical approach, we examine the stability and attractor
properties of the solutions. As we show, the first constant-roll era is
dynamically unstable towards linear perturbations and the cosmological system
is driven by the attractor solution to the final constant-roll era. As we
demonstrate, it is possible to have a nearly-scale invariant power spectrum of
primordial curvature perturbations in some cases, however this is strongly
model dependent and depends on the rate of the final constant-roll era.
Finally, we present in brief the essential features of a model that allows
oscillations between constant-roll eras.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory