Applications of the theories of dynamical systems in f(R)gravitation

In this study, we will consider the 𝑓(𝑅) theory of gravity, which is called the modified gravity theory, which is put forward outside the General Theory of Relativity (GRT) to explain the accelerating expansion of the universe without giving any components containing dark energy as a result of the calculations we will make. Our aim is to determine the cosmological evolution of the 𝑓(𝑅) theory of gravity, which is one of the modified gravitational theories, and what kind of changes and differences it brings with respect to the standard cosmology of the General Theory of Relativity. By using the 1+3 space-time covariant method, we will introduce the necessary mathematical and physical concepts in the RW (Robertson Walker) model, which includes an orthogonal and one-component fluid, and will establish and present a systematic and detailed inference of the evolution and bond equations within the framework of our own sign and symbol assumptions in the theory of gravity. We will also show that evolutionary equations can be a system of equations consisting of ordinary differential equations in terms of first-order and dimensionless variables. These types of theories have attracted a lot of attention over the past decade to fill the deficiencies of the General Theory of Relativity (GRT) in standard cosmology. However, adding higher order curvature corrections to field equations leads to much more complex and higher non-linear equations, thus making it difficult to find exact analytical solutions. Here, we want to determine by using the dynamic system theory used in various branches of science (economics, biology, etc.) by making qualitative analysis.