Fixed point analysis with steep exponential potential

Abstract

We investigate the dynamics of the steep exponential cosmological model by recasting it as a
Newtonian-type dynamical system. Our analysis focuses on the autonomous dynamical
system with particular attention to the asymptotic behavior of the  cosmological model. We
introduce a set of first-order differential equations governing the evolution of the variables.
We identify the critical points of this system and examine their stability using eigenvalue
analysis. The classification of these points—stable, unstable, or saddle—provides insight into
the qualitative evolution of the universe. The dynamics of the scalar field, characterized by a
potential V(ϕ), are analyzed using dimensionless variables within an autonomous system
framework. We find that the steep exponential potential yields a stable critical point for
certain parameter ranges, representing a late-time attractor. Using phase-space analysis, we
visualize the trajectories and stability properties of the model, offering a comprehensive
picture of scalar-field-driven cosmological evolution.