Chaotic vibrations of a nonlinear viscoelastic beam

The differential equation of motion of a nonlinear viscoelastic beam is established and is based on a novel and sophisticated stress-strain law for polymers. Applying this equation we examine a periodically forced oscillation of such a simply supported beam and search for possible chaotic responses. To this purpose we establish the Holmes-Melnikov boundary for the system. All further investigations are developed by means of a computer simulation. In this connection the authors examine critically the Poincaré mapping and the Lyapunov exponent techniques and distinguish in this way between chaotic and regular motion, A set of control parameters of the equation is found, for which either a chaotic or a regular motion can be generated, depending on the initial conditions and the corresponding basins of attraction. Thus, in this particular case two attractors of completely different nature—regular and chaotic, respectively—coexist in the phase space. The basins of attraction of the two attractors for a fixed instant of time are plotted, and appear to possess a very complex fractal geometry.