  Scalar tensor theories of gravity can be formulated in the Einstein or in the
Jordan frame, which are related by the conformal transformations. Although the
two frames are describe the same physics, and are equivalent, the stability of
the field equations in two frames are not the same. Here we implement dynamical
system and phase space approach as a robustness tool to investigate this issue.
We concentrate on the Brans Dicke theory, but the results can easily be
generalized. Our analysis show that while there is one-to-one correspondence
between critical points in two frames and each critical point in one frame is
mapped to its corresponds in other frame , however stability of a critical
points in one frame does not grantee the stability in other frame. Hence an
unstable point in one frame may be mapped to a stable point in other frame. All
trajectories between two critical points in phase space in one frame are
different from their corresponds in other ones. This indicates that the
dynamical behavior of variables and cosmological parameters are different in
two frames. Hence for those features of the study which focus on observational
measurements we must use the (JF) where experimental data have their usual
interpretation
