Pulsar X-ray radiation profiles are reduced to an asymmetric Hamiltonian system in order to construct a theory of non-linear pulsations. The potential field of the system is approximated through the Taylor polynomials of the fourth order, which model three typical solutions of non-linear systems: subcritical periodic pulsations, critical aperiodic pulsations and supercritical periodic oscillations. The phase trajectories of the pulsar are computed symbolically using series solutions of the dynamic equation in even powers of the trigonometric cosine with the mathematical and scientific software Maple^TM. The series solutions are compared with numerical solutions which are computed using the Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant. It is shown that the series solutions are superior to the numerical ones due to their uniform convergence versus a local convergence of the numerical solutions. The symbolic solutions are qualitatively matched to the pulse profiles of the Geminga pulsar, not taking into account multiple modes and stochastic perturbations. The phase potential energy of the Geminga pulsar is approximated from the experimental results as well. Significance of the obtained results, possible theoretical developments, and future applications are discussed.