  We study two transition periods in cosmology: radiation-to-matter and
matter-to-dark energy. In each case, we define a new parameter $\chi$ given by
the ratios of the two energy densities involved in the transition. Our study of
the second epoch is motivated by the need to understand cosmic acceleration.
Assuming a dynamic dark energy is the driving force for cosmic acceleration, we
formulate a new equation of state for the dark energy given in terms of the
ratio $\chi$ and the deceleration parameter, $q$. We have analysed the
resultant system of equations, where we vary different parameters and examine
the effect on the universe's evolution. For cosmic acceleration to occur, the
EoS of the dynamic dark energy must lie in the interval $\omega_{DDE}<-2/3$ at
matter-dynamic dark energy equality( equivalently $\omega_{DDE}<-0.47$ today)
