bump, while the mode with $k^{-1}<k_*^{-1}$ will be changed by the bump according to the well known result for cyclic cosmologies. A special situation belonging to this case is the limiting case $\Delta \rightarrow 0$.

We divide the evolution of fluctuation modes into five regions as shown on Fig. \[bounceplot\] [^4]. The five regions are denoted by Region I, Region II, Region III, Region IV and Region V, respectively. Region I and Region II are separated by the time $-\tau_B$. Region IV and Region V are separated by $\tau_B$. The separation between Regions II and III, and between Regions III and IV are more complicated. Because of the existence of the flat plateau (or of the local maximum of the comoving Hubble radius in the case of Model 2), we can see that there is a clear distinction between the large scale and small scale modes separated by a characteristic scale $k_*^{-1}$. Small scale modes (i.e. $k^{-1}<k_*^{-1}$) enter the Hubble radius at time $-\tau_H(k)
\leq - \Delta$ and exit the Hubble radius at $\tau_H(k) > \Delta$, and the separation between Regions II and III and between Regions III and IV are given by $-\tau_H(k)$ and $\tau_H(k)$, respectively. For large scale 