Danil Lagunov

This paper considers the motion of an elliptic foil in the presence of point vortices. For the case of a vortex pair, a bifurcation analysis of the relative equilibria (a generalization of Föppl solutions) is carried out. These solutions correspond to the motion of the system on an invariant manifold on which the dynamics is governed by a system with $\frac{3}{2}$ degrees of freedom. Using a period advance Poincaré map, a numerical analysis of the dynamics on the invariant manifold is performed for the case where the center of mass of the system moves periodically in an impulse-like manner. The occurrence of new periodic, quasiperiodic and chaotic modes of motion is demonstrated.

This paper is concerned with the motion of an elliptic foil in the field of a fixed point singularity. A complex potential of the fluid flow is constructed, and the forces and the torque which act on the foil from the fluid are obtained. It is shown that the equations of motion of the elliptic foil in the field of a fixed point vortex source can be represented as Lagrange – Euler equations. It is also shown that the system has an additional first integral due to the conservation of the angular momentum. An effective potential of the system under consideration is constructed. For the cases where the singularity is a vortex or a source, unstable relative equilibrium points corresponding to the circular motion of the foil around the singularity are found.