  Contracting Universe (including bouncing models) solution generally depends
on the initial conditions and hence possesses an extreme fine-tuning problem.
In order to probe the stability of those solutions, in this work, we consider
the Brans-Dicke theory with the power law potential in the presence of an
additional barotropic matter in the homogeneous and isotropic background. We
study the phase space and obtain critical points. We find that the quadratic
potential is a special case where one of the solutions always gives rise to
de-Sitter solution in all conformally connected frame. We generalize the
condition for arbitrary power law potential and find that contracting Universe
solution can indeed lead to an attractor solution. In doing so, we also provide
an example of a matter contracting Universe that leads to near scale-invariant
spectra and study the behavior near the fixed point. In the vicinity of this
point, the system behaves as the Universe contains three different types of
matter. Amongst them, with time, energy densities of two effective fluids decay
down and only the leading order solution survives. Therefore, the non-minimal
coupling can address and solve the problem of fine-tuning of the contraction
models and may open a different outlook.
