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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 $$ \epsilon^2(a^2(t)y')^{'}+f(y)=m(t), \quad 0<\epsilon\ll 1,$$ where $a(\cdot), m(\cdot)$ are $C^1$-functions on a given interval and $f(\cdot)$ is a $C^1$-function on $\Bbb R$. 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.
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
Full Text: DOI
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