  We study cosmological braneworld models with a single timelike extra
dimension. Such models admit the intriguing possibility that a contracting
braneworld experiences a natural bounce without ever reaching a singular state.
This feature persists in the case of anisotropic braneworlds under some
additional and not very restrictive assumptions. Generalizing our study to
braneworld models containing an induced brane curvature term, we find that a
FRW-type singularity is once again absent if the bulk extra dimension is
timelike. In this case, the universe either has a non-singular origin or
commences its expansion from a quasi-singular state during which both the
Hubble parameter and the energy density and pressure remain finite while the
curvature tensor diverges. The non-singular and quasi-singular behaviour which
we have discovered differs both qualitatively and quantitatively from what is
usually observed in braneworld models with spacelike extra dimensions and could
have interesting cosmological implications.
