Motion of a falling card in a fluid: Finite-dimensional models, complex phenomena, and nonlinear dynamics

Results are reviewed relating to the planar problem for the falling card in a resisting medium based on models represented by ordinary differential equations for a small number of variables. We introduce a unified model, which gives an opportunity to conduct a comparative analysis of dynamic behaviors of models of Kozlov, Tanabe – Kaneko, Belmonte – Eisenberg – Moses and Andersen – Pesavento – Wang using common dimensionless variables and parameters. It is shown that the overall structure of the parameter spaces for the different models shows certain similarities caused obviously by the same inherent symmetry and by universal nature of the involved phenomena of nonlinear dynamics (fixed points, limit cycles, attractors, bifurcations). In concern of motion of a body of elliptical profile in a viscous medium with imposed circulation of the velocity vector and with the applied constant torque, a presence of the Lorenz-type strange attractor is discovered in the three-dimensional space of generalized velocities.