  Modified gravity theories have received increased attention lately to
understand the late time acceleration of the universe. This viewpoint
essentially modifies the geometric components of the universe. Among numerous
extension to Einstein's theory of gravity, theories which include higher order
curvature invariant, and specifically the class of $f(R)$ theories, have
received several acknowledgments. In our current work we try to understand the
late time acceleration of the universe by modifying the geometry of the space
and using dynamical system analysis. The use of this technique allows to
understand the behavior of the universe under several circumstances. Apart from
that we study the stability properties of the critical point and acceleration
phase of the universe which could then be analyzed with observational data. We
consider a particular model $f(R) = R - \mu R_{c}(R/R_{c})^{p}$ with $ 0 < p <
1, \mu, R_{c} > 0$ for the study. As a first case we consider the matter and
radiation component of the universe with an assumption of no interaction
between them. Later, as a second case we take matter, radiation and dark energy
(cosmological constant) where study on effects of linear, non-linear and no
interaction between matter and dark energy is considered and results have been
discussed in detail.
