In this article the Sagdeev's pseudopotential approach is used to study the propagation of large-amplitude ion acoustic waves in degenerate electron–positron–ion plasma. The electrons and positrons energies are taken in the ultra-relativistic region. Periodic, rarefactive and compressive solitary and super-periodic waves are predicted in the plasma system. All probable phase plots, including super-periodic, periodic, homoclinic trajectories are obtained using the bifurcation analysis of dynamical system. The effects of phase speed of ion acoustic traveling wave, chemical potential and the ratio of the unperturbed number density of protons to the unperturbed number density of electrons on the characteristic properties of the periodic wave, super periodic wave and the Sagdeev pseudopotential is investigated. The findings of this study may be useful for understanding the fundamental properties of nonlinear waves in dense objects like neutron stars and white dwarfs, where the chemical potentials and concentration of degenerate ultra-relativistic electrons and positrons are essential factors.