  We use the phase plane analysis technique of Madsen and Ellis to consider a
universe with a true cosmological constant as well as a cosmological "constant"
that is decaying. Time symmetric dynamics for the inflationary era allows
eternally bouncing models to occur. Allowing for scalar field dynamic
evolution, we find that if dark energy decays in the future, chaotic cyclic
universes exist provided the spatial curvature is positive. This is
particularly interesting in light of current observations which do not yet rule
out either closed universes or possible evolution of the cosmological constant.
We present only a proof of principle, with no definite claim on the physical
mechanism required for the present dark energy to decay.
