The concept of oscillatory Universe appears to be realistic and buried in the dynamic dark energy equation of state. We explore its evolutionary history under the frame work of general relativity. We observe that oscillations do not go unnoticed with
such an equation of state and that their effects persist later on in cosmic evolution. The `classical general relativity seems to retain the past history of oscillatory Universe in the form of increasing scale factor as the classical thermodynamics retains this history in the form of increasing cosmological entropy.
Among various phenomenological $Lambda$ models, a time-dependent model $dot Lambdasim H^3$ is selected here to investigate the $Lambda$-CDM cosmology. Using this model the expressions for the time-dependent equation of state parameter $omega$ and oth
er physical parameters are derived. It is shown that in $H^3$ model accelerated expansion of the Universe takes place at negative energy density, but with a positive pressure. It has also been possible to obtain the change of sign of the deceleration parameter $q$ during cosmic evolution.
We perform a deductive study of accelerating Universe and focus on the importance of variable time-dependent $Lambda$ in the Einsteins field equations under the phenomenological assumption, $Lambda =alpha H^2$ for the full physical range of $alpha$.
The relevance of variable $Lambda$ with regard to various key issues like dark matter, dark energy, geometry of the field, age of the Universe, deceleration parameter and barotropic equation of state has been trivially addressed. The deceleration parameter and the barotropic equation of state parameter obey a straight line relationship for a flat Universe described by Friedmann and Raychaudhuri equations. Both the parameters are found identical for $alpha = 1$.
Choosing the three phenomenological models of the dynamical cosmological term $Lambda$, viz., $Lambda sim (dot a/a)^2$, $Lambda sim {ddot a/a}$ and $Lambda sim rho$ where $a$ is the cosmic scale factor, it has been shown by the method of numerical an
alysis that the three models are equivalent for the flat Universe $k=0$. The evolution plots for dynamical cosmological term $Lambda$ vs. time $t$ and also the cosmic scale factor $a$ vs. $t$ are drawn here for $k=0, +1$. A qualitative analysis has been made from the plots which supports the idea of inflation and hence expanding Universe.
