The relevance of research. Oil production is one of the most important industries in the Russian economy at present time. It depends to a large extent on the level of applied geophysical information and measurement systems on the basis of physical methods of obtaining information. The acoustic method is one of the leading methods of borehole geophysics. The amounts of using this method are about 10 % of all volume of well logging. The relevance of the study is caused by the need to improve the accuracy of the data obtained using acoustic logging. The effectiveness of the acoustic logging depends on the quality of the package of elastic waves. It is detected by the acoustic logging devices. The development of a mathematical model of the element motion in flexible spherical converter and study of the frequency characteristics of elastic vibrations are very important scientific and practical problems. The aim of the research is to study the forced vibrations of a spherical transducer, which is a part of the acoustic logging. Results. The authors have constructed the initial differential equations of motion of an axisymmetric spherical shell element using the variational principles and taking into account the geometric nonlinearity in the form of the Kirchhoff-Love and developed the algorithm for solving the systems of nonlinear differential equations using the finite difference method, the matrix and Runge-Kutta methods. Nonlinear vibrations of a spherical shell were analyzed based on non-linear dynamics and the qualitative theory of differential equations. It is shown that the dents in the vicinity of certain surface lines are possible in problems of nonlinear dynamics of axisymmetric spherical shells It was found that the transition from harmonic vibrations to chaotic ones occurs according to the scenario of Ruelle-Takens-Newhouse for rigidly clamped spherical shell (excitation frequency is close to the natural frequency of the shell).