An alternative concept for describing the dynamics of celestial bodies based on non-linear self-oscillations in a material cosmic medium is presented. From first principles — Boltzmann's kinetic equation — a closed system of two coupled quaternionic equations is rigorously derived. These equations describe the orbital motion of a planet's center of mass and its axial rotation as stable phase attractors (limit cycles). Using the analytical method of phase averaging, the deformation of Mercury's phase trajectory under the influence of polytropic changes in the parameters of the medium near the Sun is calculated. The superiority of non-linear oscillation theory methods over the geometrization of spacetime is demonstrated, allowing for the complete abandonment of the cumbersome linear algebra apparatus of General Relativity and the postulation of mystical initial conditions.planetary dynamics, self-oscillations, Boltzmann kinetic equation, quaternion, limit cycle, phase averaging, cosmic medium, Ether, perihelion precession, general relativity alternative