Numerical continuation of solution at singular points of codimension one

S. D. Krasnikov   ; E. B. Kuznetsov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 11, pp. 1835-1856

Résumé

Numerical continuation of a solution through some singular points of the curve of solutions to algebraic or transcendental equations with a parameter is considered. Singular points of codimension one are investigated. An algorithm for constructing all the branches of the curve at a simple bifurcation point is proposed. A special regularization that allows one to pass simple cusp points as limit points is obtained. For the regularized simple cusp point, a bound on the norm of the inverse Jacobian matrix in a neighborhood of this point is found. Using this bound, the convergence of the continuation process in a neighborhood of the simple cusp point is proved; an algorithm for the discrete continuation of the solution at the singular point along a smooth curve is obtained and its validity is proved. Based on a unified approach, a bound on the norm of the inverse Jacobian matrix and results on the convergence of continuation process in the case of the simple bifurcation point are also obtained. The operation of computational programs is demonstrated on benchmarks, which proves their effectiveness and confirms theoretical results. The effectiveness of software is investigated by solving the applied problem of three-rod truss stability.

Bibliographie

[1] Kuznetsov E. B., “Kriterialnyi podkhod pri issledovanii dinamicheskogo proschelkivaniya paneli”, Izv. RAN. Mekhanika tverd. tela, 1996, no. 1, 150–160

[2] Kuznetsov E. B., “Optimal parametrization in numerical construction of curve”, J. Franklin Institute, 344 (2007), 658–671 | DOI | MR | Zbl

[3] Krasnikov S. D., Kuznetsov E. B., “Chislennoe prodolzhenie resheniya v tochkakh bifurkatsii matematicheskikh modelei”, Matem. modelirovanie, 21:12 (2009), 47–58 | Zbl

[4] Krasnikov S. D., Kuznetsov E. B., “Parametrizatsiya resheniya v tochkakh bifurkatsii”, Differents. ur-niya, 45:8 (2009), 1194–1198 | MR | Zbl

[5] Shalashilin V. I., Kuznetsov E. B., Metod prodolzheniya resheniya po parametru i nailuchshaya parametrizatsiya v prikladnoi matematike i mekhanike, Editorial URSS, M., 1999 | MR

[6] Shalashilin V. I., Kuznetsov E. B., “Nailuchshii parametr prodolzheniya resheniya”, Dokl. AN, 334:5 (1994), 566–568 | Zbl

[7] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 2002 | MR

[8] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh integralnykh uravnenii, Nauka, M., 1969 | MR

[9] Keller H. B., Lectures on numerical methods in bifurcation problems, Springer, Berlin, 1987 | MR

[10] Allgower E. L., Georg K., Numerical continuation methods: an introduction, Series in Computational Mathematics, 13, Springer-Verlag, Berlin–Heidelberg–New York, 1990 | DOI | MR | Zbl

[11] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, GITTL, M., 1956, 392 pp. | MR

[12] Arnold V. I., Afraimovich V. S., Ilyashenko Yu. S., Shilnikov L. P., “Teoriya bifurkatsii”, Dinamicheskie sistemy – 5, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 5, VINITI, M., 1986, 5–218

[13] Crawford J. D., “Introduction to bifurcation theory”, Reviews of Modern Phys., 63:4 (1991), 991–1037 | DOI | MR

[14] Grigolyuk E. I., Lopanitsyn E. A., Konechnye progiby, ustoichivost i zakriticheskoe povedenie tonkikh pologikh obolochek, Izd-vo MAMI, M., 2004

[15] Kielhofer H., Bifurcation theory: an introduction with applications to PDEs, Appl. Math. Sciences, 156, Springer-Verlag, New York, 2004 | DOI | MR

[16] Kuznetsov E. B., Shalashilin V. I., “Parametricheskoe priblizhenie”, Zh. vychisl. matem. i matem. fiz., 34:12 (1994), 1757–1769 | MR | Zbl

[17] Kuznetsov E. B., Yakimovich A. Yu., “Nailuchshaya parametrizatsiya v zadachakh priblizheniya krivykh i poverkhnostei”, Zh. vychisl. matem. i matem. fiz., 45:5 (2005), 760–774 | MR | Zbl

[18] Krasnikov S. D., Kuznetsov E. B., “Parametrizatsiya chislennogo resheniya kraevykh zadach dlya nelineinykh differentsialnykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 45:2 (2005), 2148–2158 | MR | Zbl

[19] Kuznetsov E. B., “O nailuchshei parametrizatsii”, Zh. vychisl. matem. i matem. fiz., 48:12 (2008), 2129–2140 | MR

[20] Kuznetsov E. B., “Mnogomernaya parametrizatsiya i chislennoe reshenie sistem nelineinykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 50:2 (2010), 255–267 | MR | Zbl

[21] Kuznetsov E. B., “Prodolzhenie resheniya v mnogoparametricheskikh zadachakh priblizheniya krivykh i poverkhnostei”, Zh. vychisl. matem. i matem. fiz., 52:8 (2012), 1457–1471 | MR | Zbl

[22] Zorich V. A., Matematicheskii analiz, v. 1, Fazis, M., 1997 | MR

[23] Crandall M. G., Rabinowitz P. H., “Bifurcation from simple eigenvalues”, J. Funct. Anal., 8:2 (1971), 321–340 | DOI | MR | Zbl

[24] Kakhaner D., Mouler K., Nesh S., Chislennye metody i programmnoe obespechenie, Mir, M., 1998

[25] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, v. 1, Nauka, M., 1969

[26] Trenogin V. A., “Computation of one-parametric families of solutions of nonlinear equations”, Proc. of the Second ISAAK Congress, v. 1, 2000, 727–735 | DOI | MR | Zbl

[27] Kuznetsov E. B., “Nailuchshaya parametrizatsiya pri postroenii krivykh”, Zh. vychisl. matem. i matem. fiz., 44:9 (2004), 1540–1551 | MR | Zbl

[28] Decker D. W., Keller H. B., “Path following near bifurcation”, CPAM, XXXIV (1981), 149–175 | MR | Zbl

[29] Ginzburg V. L., Landau L. D., “K teorii sverkhprovodimosti”, Zh. eksperim. i teor. fiz., 20:12 (1950), 1064–1082

[30] Grigolyuk E. I., Shalashilin V. I., Problemy nelineinogo deformirovaniya, Nauka, M., 1988 | MR