Geodesic
Finding bifurcations for solutions of nonlinear equations by quadratic programming methods
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 3, pp. 350-364 Cet article a éte moissonné depuis la source Math-Net.Ru

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The authors study solution to nonlinear equations bifurcation of the turn point type. Here method of extended functional is used and bifurcation is found as solution to variational problems of minimax type. Iteration algorithm is constructed on the ground of the steepest descend for the piecewise-smooth maps. 
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A. A. Ivanov; Ya. Sh. Il'yasov. Finding bifurcations for solutions of nonlinear equations by quadratic programming methods. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 3, pp. 350-364. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_3_a4/

[1] Doedel E. J., Keller H. V., Kernevez J. P., “Numerical analysis and control of bifurcation problems: Bifurcation in infinite dimensions”, Int. J. Bifurcation and Chaos, 1:4 (1991), 745–772 | DOI | MR | Zbl

[2] Keller H. B., “Numerical solution of bifurcation and nonlinear eigenvalue problems”, Appl. bifurcation theory, Proc. adv. Semin. (Madison/Wis. 1976), 1977, 359–384 | MR | Zbl

[3] Seydel R., Practical bifurcation and stability analysis, Interdisciplinary Applied Mathematics, 5, 3rd ed., Springer, New York, 2009 | MR

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

[5] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhizdat, M., 1956 | MR

[6] Rabinowitz P. H., “Some global results for nonlinear eigenvalue problems”, J. Funct. Anal., 1971, no. 7, 487–513 | DOI | MR | Zbl

[7] Nirenberg L., Lektsii po nelineinomu funktsionalnomu analizu, Mir, M., 1977 | MR | Zbl

[8] Dancer E. N., “On the structure of solutions of non-linear eigenvalue problems”, Ind. Univ. Math. J., 1974, no. 3, 1069–1076 | DOI | MR | Zbl

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

[10] Kuznetsov Yu. A., Elements of applied bifurcation theory, Applied mathematical sciences, 112, Springer, New York, 1995/1998 | DOI | MR | Zbl

[11] Abbott J. P., Numerical continuation methods for nonlinear equations and bifurcation problems, Ph. D. Thesis, Australian National University, Canberra, 1977

[12] Riks E., “Application of Newton's method to the problem of elastic stability”, J. Appl. Mech., Trans. ASME, 1972, no. 39, 1060–1065 | DOI | Zbl

[13] Seydel R., Numerische bereehnung von verzweigungen boi gewohnlichen differentialgleichungen, Ph. D. Thesis, 1977 | Zbl

[14] Ilyasov Ya. Sh., “Ischislenie bifurkatsii metodom prodolzhennogo funktsionala”, Funkts. analiz i ego prilozh., 41:1 (2007), 23–38 | DOI | MR

[15] Il'yasov Y. Sh., Runst T., “Positive solutions of indefinite equations with p-Laplacian and supercritical nonlinearity”, Complex Var. Elliptic Equ., 56:10–11, 945–954 | MR

[16] Il'yasov Y., “A duality principle corresponding to the parabolic equations”, Physica D, 2370:50 (2008), 692–698 | DOI | MR

[17] Lubyshev V., “Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem”, Commun. Pure Appl. Anal., 8:3 (2009), 999–1018 | DOI | MR | Zbl

[18] Demyanov V. F., Malozemov V. N., Vvedenie v minimaks, Nauka, M., 1972 | MR

[19] Demyanov V. F., Stavroulakis G. E., Polyakova L. N., Panagiotopoulos P. D., Quasidiffercntiability and nonsmooth modelling in mechanics, engineering and economics, Kluwer, London, 1996 | MR

[20] Cabre X., “Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems”, Contemporary Math., 446, 2007, 159–174 | DOI | MR

[21] Bertsekas D. P., Nonlinear programming, II-d ed., Athena Scientific, 1999 | Zbl

[22] Fletcher R., Practical methods of optimization, New York, 2000 | MR

[23] Ambrosetti A., Brezis H., Cerami G., “Combined effect of concave and convex nonlinearities in some elliptic problems”, J. Funct. Anal., 122 (1994), 519–543 | DOI | MR | Zbl

[24] Godunov S. K., Ryabenkii V. S., Vvedenie v teoriyu raznostnykh skhem, Fizmatgiz, M., 1962 | MR