We present a comprehensive dynamical systems analysis of homogeneous and
isotropic Friedmann-La\^{i}matre-Robertson-Walker cosmologies in the Hu-Sawicki
$f(R)$ dark energy model for the parameter choice $\{n,{C}_{1}\}=\{1,1\}$ . For a
generic $f(R)$ theory, we outline the procedures of compactification of the
phase space, which in general is 4-dimensional. We also outline how, given an
$f(R)$ model, one can determine the coordinate of the phase space point that
corresponds to the present day universe and the equation of a surface in the
phase space that represents the $\mathrm{\Lambda}$ CDM evolution history. Next, we apply
these procedures to the Hu-Sawicki model under consideration. We identify some
novel features of the phase space of the model such as the existence of
invariant submanifolds and 2-dimensional sheets of fixed points. We determine
the physically viable region of the phase space, the fixed point corresponding
to possible matter dominated epochs and discuss the possibility of a
non-singular bounce, re-collapse and cyclic evolution. We also provide a
numerical analysis comparing the $\mathrm{\Lambda}$ CDM evolution and the Hu-Sawicki
evolution.