Dynamical System Perspective of Cosmological Models Minimally Coupled with Scalar Field

The stability criteria for the dynamical system of a homogeneous and isotropic cosmological model are investigated with the
interaction of a scalar field in the presence of a perfect fluid. In this paper, we depict the dynamical system perspective to study
qualitatively the scalar field cosmology under two special cases, with and without potential. In the absence of potential, we get a
two-dimensional dynamical system, and we study the analytical as well as geometrical behavior. For the dynamical system with
potential, we analyze different potential forms: simple exponential potential form (VðϕÞ = Voe−λϕ), double exponential potential
form VðϕÞ = Vo exp ð−A exp ð √2 p αϕÞÞ, and inverse power law potential form (VðϕÞ = Voϕ−α). We generate an autonomous
system of ordinary differential equations (ASODE) for each case by introducing new dimensionless variables and obtain
respective fixed points. We also analyze the type, nature, and stability of the fixed points and how their behavior reflects towards
the cosmological scenarios. Throughout the whole work, the investigation of this model has shown us the deep connection
between these theories and cosmic acceleration phenomena. The phase plots of the system at different conditions and different
values of γ have been analyzed in detail, and their geometrical interpretations have been studied. The perturbation plots of the
dynamical system have been analyzed with emphasis on our analytical findings. We have evaluated the total energy density (Ωϕ)
at the fixed points and also found out the suitable range of γ and λ for a stable model.