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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 \(\mathbb R\). 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 and coupled oscillators for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A26 Geometric methods in ordinary differential equations
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[1] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equations 31 (1979), 53–98. · Zbl 0476.34034
[2] V. Gaitsgory and G. Grammel, On the construction of asymptotically optimal controls for singularly perturbed systems. Systems Control Lett. 30 (1997), Nos. 2–3, 139–147. · Zbl 0901.93042
[3] F. Herzel and B. Heinemann, High-frequency noise of bipolar devices in consideration of carrier heating and low temperature effects. Solid-State Electronics 38 (1995), No. 11, 1905–1909.
[4] C. Jones, Geometric singular perturbation theory. Lect. Notes Math. 1609, Springer-Verlag, Heidelberg (1995). · Zbl 0840.58040
[5] M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion. J. Differ. Equations 174 (2001), No. 2, 312–368. · Zbl 0994.34032
[6] N. M. Krylov and N. N. Bogoliubov, Introduction to nonlinear mechanics. Princeton Univ. Press (1947). · Zbl 0063.03382
[7] P. Mei, C. Cai, and Y. Zou, A generalized KYP lemma-based approach for H control of singularly perturbed systems. Circuits Systems Signal Process. 28 (2009), No. 6, 945–957. · Zbl 1191.93029
[8] J. Sanders, F. Verhulst, and J. Murdock, Averaging methods in nonlinear dynamical systems. Springer-Verlag, New York (2007). · Zbl 1128.34001
[9] R. Srebro, The Duffing oscillator: a model for the dynamics of the neuronal groups comprising the transient evoked potential. Electroencephalography Clinical Neurophysiology, 96 (1995), No. 6, 561–573.
[10] B. S. Wua, W. P. Suna, and C. W. Lim, Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations. J. Sound Vibration 307 (2007), Nos. 3–5, 953–960.
[11] Y. Ye, L. Yue, D. P. Mandic, and Y. Bao-Jun, Regular nonlinear response of the driven Duffing oscillator to chaotic time series. Chinese Phys. B 18 958, (2009); doi: 10.1088/1674-1056/18/3/020 .
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