Modified gravity theories have received increased attention lately to understand the late time acceleration of the universe. This viewpoint essentially modifies the geometric components of the universe. Among numerous extension to Einstein's theory of gravity, theories which include higher order curvature invariant, and specifically the class of f (R) theories, have received several acknowledgments. In our current work we try to understand the late time acceleration of the universe by modifying the geometry of the space and using dynamical system analysis. The use of this technique allows to understand the behavior of the universe under several circumstances. Apart from that we study the stability properties of the critical point and acceleration phase of the universe which could then be analyzed with observational data. We consider a particular model f (R) = R − µR c (R/R c ) p with 0 < p < 1, µ, R c > 0 for the study. As a first case we consider the matter and radiation component of the universe with an assumption of no interaction between them. Later, as a second case we take matter, radiation and dark energy (cosmological constant) where study on effects of linear, non-linear and no interaction between matter and dark energy is considered and results have been discussed in detail.

The authors considered the bulk viscous fluid in f (R, T ) gravity within the framework of Kaluza-Klein space time. The bulk viscous coefficient (ξ) expressed as ξ = ξ 0 + ξ 1ȧ a + ξ 2ä a , where ξ 0 , ξ 1 and ξ 2 are positive constants. We take p = (γ − 1)ρ, where 0 ≤ γ ≤ 2 as an equation of state for perfect fluid. The exact solutions to the corresponding field equations are given by assuming a particular model of the form of f (R, T ) = R + 2f (T ), where f (T ) = λT , λ is constant. We studied the cosmological model in two stages, in first stage: we studied the model with no viscosity, and in second stage: we studied the model involve with viscosity. The cosmological model involve with viscosity is studied by five possible scenarios for bulk viscous fluid coefficient (ξ). The total bulk viscous coefficient seems to be negative, when the bulk viscous coefficient is proportional to ξ 2ä a , hence the second law of thermodynamics is not valid, however, it is valid with the generalized second law of thermodynamics. The total bulk viscous coefficient seems to be positive, when, the bulk viscous coefficient is proportional to ξ = ξ 1ȧ a , ξ = ξ 1ȧ a + ξ 2ä a and ξ = ξ 0 + ξ 1ȧ a + ξ 2ä a , so the second law of thermodynamics and the generalized second law of thermodynamics is satisfied throughout the evolution. We calculate statefinder parameters of the model and observed that, it is different from the ∧CDM model. Finally, some physical and geometrical properties of the models are discussed.where h.c. means Hermitian conjugate, S E is the Einstein action describing gravity, E = det(−g µν ) 1 2 , α is the gravitational coupling constant and R is the Ricci scalar.The original Kaluza-Klein theory derive with one extra spatial dimension. The appropriate metric tensor for five dimensional space time iŝThe fifth dimension is postulated to be comfactified, rolled-up in a small circle, which provides us the explanation for the un-observability of the extra dimension. Hence the topology of the five dimensional space time is M 4 × S 1 , where M 4 is the standard four-dimensional Minkowski spacetime and S 1 is a circle with very small radius. The simplest way to imagine space with one extra dimension is to imagine a small circle at every point of 3-dimensional space.Inflation is an important idea in cosmology. There are two scenarios proposed in Kaluza-Klein cosmology. The first scenario [3] is: the scale of standard 3-dimensional space expands when the scale of internal space changes slowly with time. The second scenario ([4], [5]) is: inflation occurs near the singularity a(t) → ∞, b(t) → 0 (t → t 0 ).Expansion of our universe is in an accelerating way which is suggested by type Ia supernova observational data ([6], [7]). Myrzakulov [8] constructed several concrete models describing the trefoil and figure-eight knot universes from Bianchi-type I cosmology and examined the cosmological features and properties in detail. Yesmakhanova et al [9] constructed a cosmological model by assuming the periodic forms for pressure and en...

The late time accelerated expansion of the universe can be realized using scalar fields with given self-interacting potentials. Here we consider a straightforward approach where a three cosmic fluid mixture is assumed. The fluids are standard matter perfect fluid, dark matter, and a scalar field with the role of dark energy. A dynamical system analysis is developed in this context. A central role is played by the equation of state ω ef f which determines the acceleration phase of the models. Determining the domination of a particular fluid at certain stages of the universe history by stability analysis allows, in principle, to establish the succession of the various cosmological eras.

In this work we try to understand the late-time acceleration of the universe by assuming some modification in the geometry of the space and using dynamical system analysis. This technique allows to understand the behavior of the universe without analytically solving the field equations. We study the acceleration phase of the universe and stability properties of the critical points which could be compared with observational results. We consider an asymptotic behavior of two particular models [Formula: see text] and [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text] for the study. As a first case we fix the value of [Formula: see text] and analyze for all [Formula: see text]. Later as second case, we fix the value of [Formula: see text] and calculation are done for all [Formula: see text]. At the end all the calculations for the generalized case have been shown and results have been discussed in detail.

We explore an autonomous system analysis of dark energy models with interactions between dark energy and cold dark matter in a general systematic approach to cosmological fluids. We investigate two types of models such as local and non-local ones. In particular, a local form of interaction is directly proportional to only the energy density, while a non-local interaction is directly proportional to the energy density as well as the Hubble parameter. As a consequence, it is explicitly demonstrated that in both cases there exist the stability points in terms of cosmological parameters. This work aims at obtaining acceleration and stability using interaction models without modifying the matter or geometric component of the Universe.