On the motion of a material point on a fixed ellipsoidal surface

The nonlinear dynamics of a point that remains throughout its motion on the inner part of an absolutely smooth surface of a fixed triaxial ellipsoid is studied. The motion occurs in a uniform field of gravity, the largest of the axes of the ellipsoid is directed along the vertical. The main attention is paid to the motions of the point near its stable equilibrium position at the lowest point of the ellipsoid‘s surface lying on its vertical axis. A qualitative description of conditionally periodic oscillations of the point is given, and an estimate of the measure of the set of initial conditions corresponding to these oscillations is defined. In the resonant case, when the ratio of the frequencies of small linear oscillations is equal to two, the periodic motions of the point are studied; the question of their existence, stability and geometric representation is considered.