A new class of inflationary models, that generalizes the constant roll constraint

Recently a class of inflationary models satisfying the constant rate of roll constraint, $\ddot{\phi}+(3+\alpha)H\dot{\phi}=0$, has been studied and compared with the latest cosmological observational data. We consider the broader class of constraints $\ddot{\phi}+\frac{\alpha_1}{2}\dot{\phi}+\alpha_2 H\dot{\phi}=0$ and find an exact particular solution, without initial singularity, which is an attractor of the dynamics governed by a positive periodic potential $V(\phi)=E+ F \cos (\Omega \phi )+G \cos (2 \Omega \phi )$. The spectral index $n_s$ and the tensor to scalar ratio $r$ are consistent with the recent observational data. If we ignore reheating and creation of matter after the end of inflation, this solution develops a period of deceleration and a second period of acceleration, consistent with the experimental results. In addition it yields the value of the Hubble parameter $H_0$ and the value of the redshift for the beginning of the second phase of accelerated expansion in agreement with current data.