Stability of a Class of Non-Static Axial Self-Gravitating Systems in $f(R)$ Gravity

In this paper, we analyze stability regions of a non-static restricted class of axially symmetric spacetime with anisotropic matter distribution. We consider $f(R)=R+ϵR^2$ model and assume hydrostatic equilibrium of the axial self-gravitating system at large past time. Considering perturbation from hydrostatic phase, we develop dynamical as well as collapse equations and explore dynamical instabilities at Newtonian and post-Newtonian regimes. It is concluded with the help of stiffness parameter, ${\mathrm{\Gamma }}_1$, that radial profile of physical parameters like pressure anisotropy, energy density and higher curvature terms of the $f(R)$ model affect the instability ranges.