  In bouncing cosmology, the primordial fluctuations are generated in a cosmic
contraction phase before the bounce into the current expansion phase. For a
nonsingular bounce, curvature and anisotropy grow rapidly during the bouncing
phase, raising questions about the reliability of perturbative analysis. In
this paper, we study the evolution of adiabatic perturbations in a nonsingular
bounce by nonperturbative methods including numerical simulations of the
nonsingular bounce and the covariant formalism for calculating nonlinear
perturbations. We show that the bounce is disrupted in regions of the universe
with significant inhomogeneity and anisotropy over the background energy
density, but is achieved in regions that are relatively homogeneous and
isotropic. Sufficiently small perturbations, consistent with observational
constraints, can pass through the nonsingular bounce with negligible alteration
from nonlinearity. We follow scale invariant perturbations generated in a
matter-like contraction phase through the bounce. Their amplitude in the
expansion phase is determined by the growing mode in the contraction phase, and
the scale invariance is well preserved across the bounce.
