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Resonantly forced motion of two quadratically coupled oscillators. (English) Zbl 0578.70023
Summary: The resonant response of two quadratically coupled, weakly damped oscillators with natural frequencies \(\omega_ 1\) and \(\omega_ 2,\quad \omega_ 2\approx 2\omega_ 1\), to sinusoidal forcing at frequency \(\omega\) is analyzed in the parametric domain \(\omega -\omega_ 1=O(\epsilon^{1/2}\omega)\), where \(\epsilon\) is a measure of the excitation. There is a finite neighborhood of \(\omega =\omega_ 1\), bounded by Hopf-bifurcation points, in which no stable harmonic motion is possible and the motions are either periodically modulated sinusoids (limit cycles) or chaotic. The analytical predictions are supported by numerical integration, which also reveals that the spectral domain of finite-amplitude limit cycles and chaotic motions overlaps that of stable harmonic motions. Stable harmonic motion (not necessarily unique) is possible throughout, and neither limit cycles nor chaotic motions occur in, the resonant neihborhood if the forcing is at \(2\omega\).

MSC:
70K40 Forced motions for nonlinear problems in mechanics
70K30 Nonlinear resonances for nonlinear problems in mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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