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Frequency control of singularly perturbed forced Duffing’s oscillator. (English) Zbl 1238.34071

Attention is focused on nonlinear oscillations in the context of the singularly perturbed forced oscillator of Duffing’s type with a nonlinear restoring force

ϵ 2 (a 2 (t)y ' ) ' +f(y)=m(t),0<ϵ1,

where a(·),m(·) are C 1 -functions on a given interval and f(·) is a C 1 -function on . The appearance of large frequency nonlinear oscillations of the solutions is explained. It is shown that the frequency can be controlled by a small parameter at the highest derivative. Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations are derived. A new method for the analysis of nonlinear oscillations which is based on a dynamic change of coordinates is proposed.

MSC:
34C15Nonlinear oscillations, coupled oscillators (ODE)
34E15Asymptotic singular perturbations, general theory (ODE)
34C20Transformation and reduction of ODE and systems, normal forms
34A26Geometric methods in differential equations
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