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Asymptotic theory of oscillations in Vitt systems. (English. Russian original) Zbl 1014.35057

J. Math. Sci., New York 105, No. 1, 1697-1737 (2001); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat Obz. 67, 5-68 (1999).
This paper concerns the linear system of telegraph equations \(u_t=-v_x\), \(v_t=-u_x- \varepsilon v\), \(0<x<1\), with nonlinear boundary conditions \([v-\varepsilon \alpha(u+ pu^2-{1\over 3}u^3)]_{x=0}= [pv_x+v]_{ x=1} =0\). The existence of time-periodic solutions was shown by A. Vitt [Z. Techn. Physik 4, 146-157 (1934; Zbl 0008.23301)] already in 1934. The linearized (in \(u=v=0\), \(\varepsilon=0)\) system has infinitely many imaginary eigenvalues. This is the reason that the so-called buffering phenomenon can occure: For sufficiently small \(\varepsilon\) there exist arbitrarily many stable time-periodic solutions. More exactly, the authors describe the open set in the \((\alpha,p)\)-plane such that buffering happens. The proofs are based on the so-called method of quasinormal forms, developed by Yu. S. Kolesov [Russ. Acad. Sci., Sb., Math. 78, No. 2, 367-378 (1994; Zbl 0817.35038)].

MSC:

35L50 Initial-boundary value problems for first-order hyperbolic systems
35B10 Periodic solutions to PDEs
35B32 Bifurcations in context of PDEs
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