Gadyl’shin, R. R. On the poles of an acoustic resonator. (English. Russian original) Zbl 0810.76079 Funct. Anal. Appl. 27, No. 4, 229-239 (1993); translation from Funkts. Anal. Prilozh. 27, No. 4, 3-16 (1993). We prove that under some conditions there exists a unique resonator pole, its order is equal to one, and a unique eigenfunction is associated with it. The existence of exactly two vanishing first-order poles is shown, and asymptotic expansions of these poles into power series with respect to a small parameter (the channel cross-section diameter) are constructed. We give a representation of the solution to the boundary value problem which allows us to justify asymptotic expansions of the poles and the solution. Cited in 2 Documents MSC: 76Q05 Hydro- and aero-acoustics 35P20 Asymptotic distributions of eigenvalues in context of PDEs Keywords:Neumann problem; unique resonator pole; unique eigenfunction; asymptotic expansion; small parameter × Cite Format Result Cite Review PDF References: [1] V. I. Arnold, ”Remarks on perturbation theory for problems of Mathieu type,” Usp. Mat. Nauk,38, No. 4, 189–203 (1983). [2] V. I. Arnold, ”Small denominators I. Mappings of the circumference onto itself,” Trans. Amer. Math. Soc.,46, 213–284 (1965). · Zbl 0152.41905 [3] O. G. Galkin, ”Phase-locking for Mathieu-type vector fields on the torus,” Funkts. Anal. Prilozhen.,26, No. 1, 1–8 (1992). · Zbl 0828.20025 · doi:10.1007/BF01077066 [4] A. Khinchin, Continued fractions, Groningen, Nordhorff (1963). · Zbl 0117.28601 [5] C. Baesens, J. Guckenheimer, S. Kim, and R. S. MacKay, ”Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos,” Phys. D,49, 387–475 (1991). · Zbl 0734.58036 · doi:10.1016/0167-2789(91)90155-3 [6] R. E. Ecke, J. D. Farmer, and D. K. Umberger, ”Scaling of the Arnold tongues,” Nonlinearity,2, 175–196 (1989). · Zbl 0689.58017 · doi:10.1088/0951-7715/2/2/001 [7] J. Franks and M. Misiurewicz, ”Rotation sets of toral flows,” Proc. Amer. Math. Soc.,109, 243–249 (1990). · Zbl 0701.57016 · doi:10.1090/S0002-9939-1990-1021217-5 [8] O. G. Galkin, ”Resonance regions for Mathieu type dynamical systems on a torus,” Phys. D,39, 287–298 (1989). · Zbl 0695.58025 · doi:10.1016/0167-2789(89)90011-0 [9] C. Grebogi, E. Ott, and J. A. Yorke, ”Attractors on ann-torus: quasiperiodicity versus chaos,” Phys. D,15, 354–373 (1985). · Zbl 0577.58023 · doi:10.1016/S0167-2789(85)80004-X [10] G. R. Hall, ”Resonance zones in two-parameter families of circle homeomorphisms,” SIAM J. Math. Anal.,15, 1075–1081 (1984). · Zbl 0554.58040 · doi:10.1137/0515083 [11] S. Kim, R. S. MacKay, and J. Guckenheimer, ”Resonance regions for families of torus maps,” Nonlinearity,2, 391–404 (1989). · Zbl 0678.58034 · doi:10.1088/0951-7715/2/3/001 [12] M. Misiurewicz and K. Ziemian, ”Rotation sets for maps of tori,” J. London Math. Soc.,40, 490–506 (1989). · Zbl 0663.58022 · doi:10.1112/jlms/s2-40.3.490 [13] S. Newhouse, J. Palis, and F. Takens, ”Bifurcations and stability of families of diffeomorphisms,” Publ. Math. IHES,57, 5–72 (1983). · Zbl 0518.58031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.