Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation

This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.