Resonance Capture in a System of Two Oscillators Near Equilibrium
- 33 Downloads
We consider a system of differential equations describing the interaction of two weakly coupled nonlinear oscillators. We assume that one oscillator is initially far from equilibrium, the other is near equilibrium, and their frequencies are close. We study the effect of resonance capture, when the frequencies of the coupled oscillators remain close while the oscillation energies change in time significantly; in particular, the second oscillator goes far from equilibrium. We find that the initial stage of resonance capture is described by the second Painlev´e equation. We obtain such a description in the asymptotic approximation in a small parameter corresponding to the coupling constant.
Keywordsnonlinear oscillation small parameter asymptotic approximation resonance capture
Unable to display preview. Download preview PDF.
- 3.N. N. Bogoliubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of NonLinear Oscillations [in Russian], Nauka, Moscow (1974).Google Scholar
- 11.A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, and A. S. Fokas, Painlevé Transcendents: Method of the Riemann Problem [in Russian], NITs RKhD, Moscow (2005); English transl.: A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Yu. Novokshenov Painlevé Transcendents: The Riemann–Hilbert Approach (Math. Surv. Monogr., Vol. 128), Amer. Math. Soc., Providence, R. I. (2006).Google Scholar
- 13.A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Nauka, Moscow (1989); English transl. (Transl. Math. Monogr., Vol. 102), Amer. Math. Soc., Providence, R. I. (1992).Google Scholar