Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. (English) Zbl 0969.70017

Belhaq, M.; Houssni, M.

Nonlinear Dyn. 18, No. 1, 1-24 (1999).

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.

Reviewer: Yuri N.Sankin (Ul’yanovsk)

MSC:

70K55	Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
70K40	Forced motions for nonlinear problems in mechanics
74H45	Vibrations in dynamical problems in solid mechanics
70Q05	Control of mechanical systems
70K43	Quasi-periodic motions and invariant tori for nonlinear problems in mechanics

Keywords:

parametric excitation; external excitation; one-mode vibrations of elastic structure; one-degree-of-freedom system; quadratic term; cubic term; Bogolyubov-Mitropolskij method; reduced system; small parameter method; nonlinear approximations; control of chaos