The role of an exponential function of the scalar curvature in the modified gravity is analyzed .
Two models are proposed .
A toy model that complies with local and cosmological constraints and gives appropriate qualitative description of the cosmic evolution .
The trajectories in the m - r plane , given by m = - ( r + 1 ) ( \eta + r ) / r , lead to saddle matter dominant critical point ( r = -1 , m = 0 ) that can evolve towards the late time de Sitter attractor at r = -2 and 0 < m \leq 1 .
Initial conditions for the case \eta = 0.68 have proposed , showing that this toy model has an acceptable matter era and gives an approximate qualitative behavior of cosmic evolution .
A second viable model , behaves very close to \Lambda CDM at early times and can satisfy local and cosmological constraints .
It behaves as R - 2 \Lambda at R \rightarrow \infty and tends to zero at R \rightarrow 0 , containing flat spacetime solution .
The model gives viable trajectories in the ( r,m ) -plane that , as the first model , connect the matter dominated point with a de Sitter attractor at r = -2 .
The cosmic evolution of the main density parameters in this model is consistent with current observations with an equation of state very close to -1 .