  We point out a new phenomenon which seems to be generic in 4d effective
theories of scalar fields coupled to Einstein gravity, when applied to
cosmology. A lift of such theories to a Weyl-invariant extension allows one to
define classical evolution through cosmological singularities unambiguously,
and hence construct geodesically complete background spacetimes. An attractor
mechanism ensures that, at the level of the effective theory, generic solutions
undergo a big crunch/big bang transition by contracting to zero size, passing
through a brief antigravity phase, shrinking to zero size again, and
re-emerging into an expanding normal gravity phase. The result may be useful
for the construction of complete bouncing cosmologies like the cyclic model.
