I will present analytic solutions to a class of cosmological models described by a canonical scalar field minimally coupled to gravity and experiencing self interactions through a hyperbolic potential. Using models and methods of solution inspired by 2T-physics, I will show how analytic solutions can be obtained including radiation and spacial curvature. Among the analytic solutions, there are many interesting geodesically complete cyclic solutions, both singular and non-singular ones. Cyclic cosmological models provide an alternative to inflation for solving the horizon and flatness problems as well as generating scale-invariant perturbations. I will argue in favor of the geodesically complete solutions as being more attractive for constructing a more satisfactory model of cosmology. When geodesic completeness is imposed, it restricts models and their parameters to certain a parameter subspace, including some quantization conditions on parameters. I will explain the theoretical origin of our model from the point of view of 2T-gravity as well as from the point of view of the colliding branes scenario. If time permits, I will discuss how to associate solutions of the quantum Wheeler-deWitt equation with the classical analytic solutions, physical aspects of some of the cyclic solutions, and outline future directions.