nonlinear dynamics of a dark energy model

We investigate a nonlinear equation of state arising in a phenomenological model of dark energy. the model is motivated by modifications to quintessence dynamics and leads to a non-linear differential equation for the hubble parameter. we provide both approximate analytical methods and numerical solutions, demonstrating how nonlinearity alters the late-time acceleration of the universe. the results suggest that deviations from linearized treatments can significantly modify the effective equation of state parameter at redshifts z < 1, with potential implications for precision cosmology.