  In this paper, we analyze the modified $f(\mathcal{R})$ gravity models in
Friedmann--Lema\^{\i}tre--Robertson--Walker (FLRW) background. The actions of
bouncing cosmology are studied under consideration of different viable models
in $f(\mathcal{R})$ gravity theory that can resolve the difficulty of
singularity in standard Big-Bang cosmology. Under different viable models in
$f(\mathcal{R})$ gravity theory, the cosmological constraints are plotted in
provisions of cosmic-time, then investigated the bounce circumstance. In
addition, the red-shift parameter is used to reconstruct the modified gravity,
and compile the cosmological parameters that infer accelerated universe
expansion. Finally, the situation stability is evaluated with a sound speed
feature, which illustrates late-time stability.
