  We 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 inspired
by 2T-physics, we show how analytic solutions can be obtained in
flat/open/closed Friedmann-Robertson-Walker universes. Among the analytic
solutions, there are many interesting geodesically complete cyclic solutions in
which the universe bounces at either zero or finite sizes. When geodesic
completeness is imposed, it restricts models and their parameters to a certain
parameter subspace, including some quantization conditions on initial
conditions in the case of zero-size bounces, but no conditions on initial
conditions for the case of finite-size bounces. We 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 in the context of M-theory. We
will indicate how to associate solutions of the quantum Wheeler-deWitt equation
with our classical analytic solutions, mention some physical aspects of the
cyclic solutions, and outline future directions.
