### 4.1 Dynamical equations

We introduce the following variables
together with the density parameters
It is straightforward to derive the following equations
where is the number of e-foldings, and
From Eq. (4.68) the Ricci scalar can be expressed by . Since depends on , this means that is a function of , that is, . The CDM model, , corresponds to . Hence the quantity characterizes the deviation of the background dynamics from the CDM model. A number of authors studied cosmological dynamics for specific f (R) models [1603824882523119828072411592351279483321432].

The effective equation of state of the system is defined by

which is equivalent to . In the absence of radiation () the fixed points for the above dynamical system are
The points and are on the line in the plane.

The matter-dominated epoch ( and ) can be realized only by the point for close to 0. In the () plane this point exists around . Either the point or can be responsible for the late-time cosmic acceleration. The former is a de Sitter point () with , in which case the condition (2.11) is satisfied. The point can give rise to the accelerated expansion () provided that , or , or .

In order to analyze the stability of the above fixed points it is sufficient to consider only time-dependent linear perturbations () around them (see [170171] for the detail of such analysis). For the point the eigenvalues for the Jacobian matrix of perturbations are

where and with . In the limit that the latter two eigenvalues reduce to . For the models with , the solutions cannot remain for a long time around the point because of the divergent behavior of the eigenvalues as . The model () falls into this category. On the other hand, if , the latter two eigenvalues in Eq. (4.77) are complex with negative real parts. Then, provided that , the point corresponds to a saddle point with a damped oscillation. Hence the solutions can stay around this point for some time and finally leave for the late-time acceleration. Then the condition for the existence of the saddle matter era is
The first condition implies that viable f (R) models need to be close to the CDM model during the matter domination. This is also required for consistency with local gravity constraints, as we will see in Section 5.

The eigenvalues for the Jacobian matrix of perturbations about the point are

where . This shows that the condition for the stability of the de Sitter point is [44024325026]
The trajectories that start from the saddle matter point satisfying the condition (4.78) and then approach the stable de Sitter point satisfying the condition (4.80) are, in general, cosmologically viable.

One can also show that is stable and accelerated for (a) , , (b) , , (c) , , (d) , . Since both and are on the line , only the trajectories from to are allowed (see Figure 2). This means that only the case (a) is viable as a stable and accelerated fixed point . In this case the effective equation of state satisfies the condition .

From the above discussion the following two classes of models are cosmologically viable.

• Class A: Models that connect (, ) to ()
• Class B: Models that connect (, ) to ()

From Eq. (4.56) the viable f (R) dark energy models need to satisfy the condition , which is consistent with the above argument.