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 Poincare 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.