Geodesic
Multifrequency parametric resonance in a~non-linear wave equation
Izvestiya. Mathematics , Tome 66 (2002) no. 6, pp. 1131-1145

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We consider the boundary-value problem $$ u_{tt}+\varepsilon u_t+\biggl(1+\varepsilon\sum_{k=1}^m\alpha_k\cos 2\varphi_k\biggr)u=a^2u_{xx}-u^2u_t,\qquad u\big|_{x=0}=u\big|_{x=\pi}=0, $$ where $0\varepsilon\ll 1$, $a>0$, $\varphi_k=\sigma_kt+c_k$, $k=1,\dots,m$. We show that a suitable choice of a positive integer $m$ and real parameters $\alpha_k$, $\sigma_k$, $k=1,\dots,m$, enables us to make this problem have any prescribed number of exponentially stable time-quasiperiodic solutions bifurcating from zero. 
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     author = {A. Yu. Kolesov and N. Kh. Rozov},
     title = {Multifrequency parametric resonance in a~non-linear wave equation},
     journal = {Izvestiya. Mathematics },
     pages = {1131--1145},
     publisher = {mathdoc},
     volume = {66},
     number = {6},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2002_66_6_a2/}
}
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A. Yu. Kolesov; N. Kh. Rozov. Multifrequency parametric resonance in a~non-linear wave equation. Izvestiya. Mathematics , Tome 66 (2002) no. 6, pp. 1131-1145. http://geodesic.mathdoc.fr/item/IM2_2002_66_6_a2/