In this paper, we analyze stability regions of a non-static restricted class of axially symmetric spacetime with anisotropic matter distribution. We consider $f(R)=R+{ \epsilon}R^2$ 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, $\Gamma_1$, that radial profile of physical parameters like pressure anisotropy, energy density and higher curvature terms of the $f(R)$ model affect the instability ranges.
Categories: gr-qc