I don't know what a nonlinear spacetime is. The Einstein field equation is nonlinear in the sense that two different spacetimes, although each a solution of the equation, their sum would not be a solution.
There is a spacetime which is solution of the Einstein field equation which is believed to be chaotic:
The mixmaster universe: A chaotic Farey tale
Neil Cornish, Janna Levin http://arxiv.org/abs/gr-qc/9612066