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.

PRD accepted