Stability of non-homegeneous models and fine tuning of initial state

We apply phase space analysis to inhomogeneous cosmological model given by Lema\^itre-Tolman model. We describe some general conditions required to interpret the model stable enough and, in the present paper, apply them to two special cases: dust filled homogeneous model with and without cosmological constant. We find that such stability explaining all present astrophysical observations can not be achieved due to instabilities in phase space. This hints that non-homogeneous models are not likely to be physically viable, although any conclusive analysis requires more realistic modeling of non-homogeneous universe.