  We consider evolutions of linear fluctuations as the background Friedmann
world model goes from contracting to expanding phases through smooth and
non-singular bouncing phases. As long as the gravity dominates over the
pressure gradient in the perturbation equation the growing-mode in the
expanding phase is characterized by a conserved amplitude, we call it a C-mode.
In the spherical geometry with a pressureless medium, we show that there exists
a special gauge-invariant combination \Phi which stays constant throughout the
evolution from the big-bang to the big-crunch with the same value even after
the bounce: it characterizes the coefficient of the C-mode. We show this result
by using a bounce model where the pressure gradient term is negligible during
the bounce; this requires additional presence of an exotic matter. In such a
bounce, even in more general situations of the equation of states before and
after the bounce, the C-mode in the expanding phase is affected only by the
C-mode in the contracting phase, thus the growing mode in the contracting phase
decays away as the world model enters expanding phase. In the case the
background curvature has significant role during the bounce, the pressure
gradient term becomes important and we cannot trace C-mode in the expanding
phase to the one before the bounce. In such situations, perturbations in a
fluid bounce model show exponential instability, whereas the ones in a scalar
field bounce model show oscillatory behaviors.
