Bifurcations of resonance oscillations and optimization of the trigonometric impulse by the nonsymmetry coefficient

D. V. Kostin

Sbornik. Mathematics, Tome 207 (2016) no. 12, pp. 1709-1728

Résumé

Methods are given for the approximate calculation of a branch of a resonance oscillation when it bifurcates from a stationary point and for optimizing this branch with respect to the nonsymmetry coefficient, which is defined as the ratio between the largest and the smallest values of the amplitude. It is shown that the optimal values of the base amplitudes are the coefficients of the corresponding Fejér series. The largest value of the nonsymmetry coefficient is calculated exactly. Bibliography: 18 titles.

Détail

Keywords: smooth functional, periodic extremal, nonsymmetry coefficient, Lyapunov-Schmidt reduction.
Mots-clés : bifurcation, Fejér trigonometric series

Bibliographie

[1] S. S. Sarkisyan, L. V. Golovanov, Prognozirovanie razvitiya bolshikh sistem, Statistika, M., 1975, 192 pp.

[2] D. V. Kostin, “K voprosu optimizatsii zakriticheskogo izgiba uprugoi lopatki turbiny”, Nasosy. Turbiny. Sistemy, 2012, no. 3(4), 67–72

[3] D. V. Kostin, “Analiz izgibov uprugoi lopatki turbiny s uchetom neodnorodnosti materiala”, Nasosy. Turbiny. Sistemy, 2013, no. 3(8), 56–61

[4] B. M. Darinskii, Yu. I. Sapronov, S. L. Tsarev, “Bifurcations of extremals of Fredholm functionals”, J. Math. Sci. (N. Y.), 145:6 (2007), 5311–5453 | DOI | MR | Zbl

[5] Yu. I. Sapronov, S. L. Tsarev, “Global comparison of finite-dimensional reduction schemes in smooth variational problems”, Math. Notes, 67:5 (2000), 631–638 | DOI | DOI | MR | Zbl

[6] S. L. Tsarev, “Sravnenie konechnomernykh reduktsii v gladkikh variatsionnykh zadachakh c simmetriei”, Sovr. matem. i ee prilozheniya, 7, In-t kibernetiki AN Gruzii, Tbilisi, 2003, 86–90

[7] V. A. Kostin, D. V. Kostin, Yu. I. Sapronov, “Maxwell–Fejér polynomials and optimization of polyharmonic impulse”, Dokl. Math., 86:1 (2012), 512–514 | DOI | Zbl

[8] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo Mosk. un-ta, M., 1965, 549 pp.

[9] V. I. Arnol'd, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps, v. I, Monogr. Math., 82, The classification of critical points, caustics and wave fronts, Birkhäuser, Boston, MA, 1985, xi+382 pp. | DOI | MR | MR | Zbl | Zbl

[10] A. V. Zachepa, Yu. I. Sapronov, “O bifurkatsii ekstremalei fredgolmova funktsionala iz vyrozhdennoi tochki minimuma s osobennostyu $3$-mernoi sborki”, Tr. matem. f-ta (novaya seriya), 9, Izd-vo VGU, Voronezh, 2005, 57–71

[11] A. P. Karpova, U. V. Ladykina, Yu. I. Sapronov, “Bifurkatsionnyi analiz fredgolmovykh uravnenii s krugovoi simmetriei i ego prilozheniya”, Matem. modeli i operatornye uravneniya, 5, ch. 1, “Sozvezdie”, VGU, Voronezh, 2008, 45–90

[12] E. V. Derunova, Yu. I. Sapronov, “Application of normalized key functions in a problem of branching of periodic extremals”, Russian Math. (Iz. VUZ), 59:8 (2015), 9–18 | DOI | Zbl

[13] P. K. Suetin, Klassicheskie ortogonalnye mnogochleny, 2-e izd., Nauka, M., 1979, 415 pp. | MR | Zbl

[14] G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, rev. ed., Amer. Math. Soc., Providence, R.I., 1959, ix+421 pp. | MR | Zbl | Zbl

[15] T. Poston, I. Stewart, Catastrophe theory and its applications, Surveys and Reference Works in Mathematics, 2, Pitman, London–San Francisco, Calif.–Melbourne, 1978, xviii+491 pp. | MR | MR | Zbl | Zbl

[16] R. Gilmore, Catastrophe theory for scientists and engineers, John Wiley Sons, New York, 1981, xvii+666 pp. | MR | Zbl | Zbl

[17] B. M. Levitan, Pochti-periodicheskie funktsii, Gostekhizdat, M., 1953, 396 pp. | MR | Zbl

[18] P. K. Suetin, Nachala matematicheskoi teorii antenn, Insvyazizdat, M., 2008, 228 pp.