In (T. Ma and S. Wang. Gravitational field equations and theory of dark
matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34(2):
335-366, 2014; arXiv:1206.5078v2), a new set of gravitational field equations are derived
based only on 1) the Einstein principle of general relativity, and 2) the principle
of interaction dynamics, due to the the presence of dark energy and dark matter. With
the field equations, we show that gravity can display both attractive and repulsive behavior,
and the dark matter and dark energy are just a property of gravity caused by
the nonlinear interactions of the gravitational potential g_{μv} and its dual field. Themain
objectives of this paper are two-fold. The first is to study the PID-induced cosmological
model, and to show explicitly, as addressed in (T. Ma and S. Wang, Astrophysical
dynamics and cosmology, Journal of Mathematical Study, 47(4): 305-378, 2014), that 1)
darkmatter is due to the curvature of space, and 2) dark energy corresponds to the negative
pressure generated by the dual gravitational potential in the field equations, and
maintains the stability of geometry and large scale structure of the Universe. Second,
for the gravitational field outside of a ball of centrally symmetric matter field, there
exist precisely two physical parameters dictating the two-dimensional stable manifold
of asymptotically flat space-time geometry, such that, as the distance to the center of
the ball of the matter field increases, gravity behaves as Newtonian gravity, then additional
attraction due to the curvature of space (dark matter effect), and repulsive (dark
energy effect). This also clearly demonstrates that both dark matter and dark energy
are just a property of gravity.