Cosmological Dynamics from Modified $f(R)$ Gravity in Einstein Frame
  In this paper, we investigate and analyze the cosmological dynamics of the universe, with an effect of modified $f(R)$ gravity emerging at cosmological scale. We choose the Einstein frame as a physical frame. We consider phase portraits of the universe at the late time from modified $f(R)$ gravity model. This result gives our universe an acceleration phase expansion without introducing existence of dark energy dominating our universe. 
Daris Samart