Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations
This paper studies dynamic response of a one-degree-of-freedom system with both quadratic and cubic nonlinearities, subjected to combined parametric and external excitations with incommensurate frequencies. Such model can describe one-mode vibrations of a heavy elastic structure suspended between two fixed supports at the same level and excited by a longitudinal and transversal periodic forces. From physical point of view, the quadratic term is present due to the curvature of the structure, whereas the cubic term is due to the symmetric nonlinearity of the material. By using the Bogolyubov-Mitropolskij method, the authors transform the original quasi-periodic system to a reduced system. The solution of the reduced system is then obtained by small parameter method which provides nonlinear approximations to the periodic solution. Finally, the authors construct a control of chaos by introducing a harmonic parametric component into the cubic term.
one-mode vibrations of elastic structure
scientific article; zbMATH DE number 1342558