The paper studies the nonlinear dynamics of a conducting spherical particle in the field of a charged flat capacitor subjected to an external periodic action. Particle oscillations from one wall of the capacitor to another are accompanied by charge exchange and impact interactions upon instantaneous contact of the particle with the walls. A mathematical model is presented that describes the dynamics of a particle, which (the model) is a dynamic non-autonomous essentially nonlinear system with a variable structure. The behavior of phase trajectories in one part of the phase space is described by differential equations, and in the other by algebraic relations, it is shown that in a symmetric dynamic system both symmetric trajectories of periodic modes of particle motion and asymmetric periodic modes of motion are possible. The mathematical apparatus used in the paper is a numerical-analytical apparatus of the method of point mappings of Poincare surfaces. The following are presented: parametric relations defining the coordinates of fixed points corresponding to symmetric periodic modes of particle motion, characteristic polynomial, and equations of existence and stability domain boundaries in the parameter space of a system of periodic symmetric oscillations of a particle with one impact of the latter on each wall of the capacitor. The study of more complex particle dynamics is carried out by a numerical-analytical method using the construction of bifurcation diagrams for all parameters of the dynamic system under study. This made it possible to indicate in the parameter space the existence and stability domains of all possible periodic asymmetric periodic modes of particle motion, including chaotic oscillations of the particle. A scenario of transition to chaos is indicated, which is similar to the Feigenbaum scenario–period doubling. Bifurcation boundaries of existence of point mappings are constructed and an algorithm for their construction is indicated.