Phase space of generalized Gauss-Bonnet dark energy

The generalized Gauss-Bonnet theory, introduced by Lagrangian F(R,G), has been considered as a general modified gravity for explanation of the dark energy. G is the Gauss-Bonnet invariant. For this model, we seek the situations under which the late-time behavior of the theory is the de Sitter space-time. This is done by studying the two-dimensional phase space of this theory, i.e. the R-H plane. By obtaining the conditions under which the de Sitter space-time is the stable attractor of this theory, several aspects of this problem have been investigated. It has been shown that there exist at least two classes of stable attractors: the singularities of the F(R,G), and the cases in which the model has a critical curve, instead of critical points. This curve is R=12H(2) in R-H plane. Several examples, including their numerical calculations, have been discussed.