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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

${ϵ}^{2}{\left({a}^{2}\left(t\right){y}^{\text{'}}\right)}^{{}^{\text{'}}}+f\left(y\right)=m\left(t\right),\phantom{\rule{1.em}{0ex}}0<ϵ\ll 1,$

where $a\left(·\right),m\left(·\right)$ are ${C}^{1}$-functions on a given interval and $f\left(·\right)$ is a ${C}^{1}$-function on $ℝ$. 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:
 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34E15 Asymptotic singular perturbations, general theory (ODE) 34C20 Transformation and reduction of ODE and systems, normal forms 34A26 Geometric methods in differential equations