Abstract
The purpose of this paper is to give an account of recent developments in numerical methods for the solution of bifurcation problems. For readers not too familiar with our subject we shall first summarize important applications of bifurcation and dicuss some of the basic ideas, problems and tools of bifurcation theory.
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References
Abbott, J. P.: An efficient algorithm for the determination of certain bifurcation points, J. Comp. Appl. Maths. 4 (1978), 19–27
Allgower, E., Georg, K.: Simplicial and continuation methods for approximating fixed points and solutions to systems of equations, Preprint No. 240, SFB 72, Universität Bonn 1979, to appear in SLAM Review
Amann, H.: Ljusternik-Schnirelman theory and nonlinear eigenvalue problems, Math. Ann. 199 (1972), 55–72
Anselone, P. M., Moore R. H.: An extension of the Newton-Kantorovic method for solving nonlinear equations with an application to elasticity, J. Math. Anal. Appls. 13 (1966), 476–501
Atkinson, K. E.: The numerical solution of a bifurcation problem, SIAM J. Numer. Anal. 14 (1977), 584–599
Bauer, L., Reiss, E. L.: Numerical bifurcation and secondary bifurcation, Proc. 3rd. Symp. Numer. Solut. Partial Differ. Equat., College Park 1975, pp. 443–467 (1976)
Bazley, N. W.: Approximation of operators with reproducing non-linearities, manuscripta math. 18 (1976), 353–369
Beyn, W.-J.: On discretizations of bifurcation problems, these proceedings
Bohl, E.: On two boundary value problems in nonlinear elasticity from a numerical viewpoint, in: Ansorge R., Törnig, W. (eds.): Numerical treatment of differential equations in applications, Springer Lecture Notes in Maths. 679, 1978
Bohl, E.: On the bifurcation diagram of discrete analogues of ordinary bifurcation problems, Math. Meth. Appl. Sci. 1 (1979), 566–571
Bongers, A.: Behandlung verallgemeinerter nichtlinearer Eigenwertprobleme mit Ljusternik-Schnirelmann-Theorie, Dissertation, Mainz 1979
Bouc, R., Defilippi, M., Iooss, G.: On a problem of forced nonlinear oscillations. Numerical example of bifurcation into an invariant torus, Nonlinear Analysis 2 (1978), 211–224
Crandall, M. G.: An introduction to constructive aspects of bifurcation and the implicit function theorem, in |57|
Crandall, M. G., Rabinowitz, P. H.: Bifurcation from simple eigenvalues, J. Functional Anal. 8 (1971), 321–340
Crandall, M. G., Rabinowitz, P. H.: Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180
Decker, D. W.: Some topics in bifurcation theory, PhD. thesis, Caltech, Pasadena, 1978
Demoulin, Y.-M. J., Chen, Y. M.: An iteration method of solving nonlinear eigenvalue problems, SIAM J. Numer. Anal. 28 (1975), 588–595
Drexler, F.-J.: Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen, Numer. Math. 29 (1977), 45–58
Drexler, F.-J.: A homotopy-method for the calculation of all zeros of zero-dimensional polynomial ideals, pp. 64–93 in |85|
Engl, H.: On the change of parameters in continuation methods, Preprint Univ. Linz, Math. Inst. Nr. 58 (1976)
Georg, K.: An iteration method for solving nonlinear eigenvalue problems, in: Albrecht, J., Collatz, L., Kirchgässner, K. (eds.): Constructive methods for nonlinear boundary value problems and nonlinear oscillations, ISNM 48, 38–47, Birkhäuser, Basel 1979
Georg, K.: On the convergence of an inverse iteration method for nonlinear elliptic eigenvalue problems, Numer. Math. 32 (1979)
Hermann, M.: Integrative Behandlung von Bifurkationsproblemen bei gewöhnlichen Differentialgleichungen, Forschungsergebnisse N/79/19, N/79/50, Universität Jena (1979), to appear in Wiss. Zeitschrift der FSU Jena
Hoffmann W., Küpper, T.: Punktweise Abschätzungen zur Ermittlung des Einflusses von Störtermen bei Verzweigungsproblemen, Report 78–22, Math. Institut der Universität zu Köln (1978)
Hopf, E.: Bifurcation of a periodic solution from a stationary solution of a system of differential equations, translation of Hopfs original paper (1942), in |45|
Iudovich, V. I.: The onset of auto-oscillations in a fluid, J. Appl. Math. Mech. 35(1971), 587–603
Jeggle, H.: Existence and discrete approximation of bifurcation points, pp. 157–186 in |85|
Keener, J. P., Keller, H. B.: Perturbed bifurcation theory, Arch. Rational Mech. Anal. 50(1974), 159–175
Keller, H. B.: Numerical solution of bifurcation and nonlinear eigenvalue problems, pp. 359–384 in |57|
Keller, H. B.: Constructive methods for bifurcation and nonlinear eigenvalue problems, pp. 241–251 in: Glowinski, R., Lions, J. L. (eds.): Computing methods in Applied Sciences and Engineering, 1977, I, Lecture Notes in Maths. 704, Springer, 1979
Keller, H. B., Langford, W. F.: Iterations, perturbations and multiplicities for nonlinear bifurcation problems, Arch. Rational Mech. Anal. 48 (1972), 83–108
Keller, J. B., Antman, S. (eds.): Bifurcation theory and nonlinear eigenvalue problems, Benjamin, New York 1969
Kesavan, S.: La méthode de Kikuchi appliquée aux équations de von Karman, Numer. Math. 32 (1979), 209–232
Kikuchi, F.: An iterative finite element scheme for bifurcation analysis of semi-linear elliptic equations, ISAS Report No, 542, Tokyo 1976
Kikuchi, F.: Finite element approximation of bifurcation problems, Theoretical and Applied Mechanics 26 (1976), 37–51, University of Tokyo Press
Kikuchi, F.: Numerical analysis of the finite element method applied to bifurcation problems of turning point type, ISAS Report No. 564, Tokyo 1978
Kikuchi, F.: Finite element approximations to bifurcation problems of turning point type, Theoretical and Applied Mechanics 27 (1977), 99–114, University of Tokyo Press
Kikuchi, F.: Finite element approximations to bifurcation problems of turning point type, pp. 243–266 in: Glowinski, R., Lions J.L. (eds.): Computing methods in Applied Sciences and Engineering, Lecture Notes in Maths. 704, Springer-Verlag, 1979
Krasnosel’skii, M. A.: Topological methods in the theory of nonlinear integral equations, Pergamon Press, Oxford 1964
Kubicek, M.: Evaluation of branching-points for nonlinear boundary-value problems based on the GPM technique, Appl. Maths. Comp. 1(1975), 341–352
Kubicek, M., Marek, M.: Evaluation of limit and bifurcation points for algebraic equations and nonlinear boundary value problems, Appl. Maths. Comp. 5 (1979), 253–264
Langford, W. F.: A shooting algorithm for the best least squares solution of two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977), 527–542
Langford, W. F.: Numerical solution of bifurcation problems for ordinary differential equations, Numer. Math. 28 (1977), 171–190
Leray, J., Schauder, J. P.: Topologie et equations fonctionelles, Ann. Ecole Norm. Sup. (3) 51 (1934), 45–78
Marsden, J. E., McCracken, M.: The Hopf bifurcation and its applications, Springer-Verlag, New York 1976
Meyer-Spasche, R.: Numerical treatment of Dirichlet problems with several solutions, in: Albrecht, J., Collatz, L. (eds.): Numerische Behandlung von Differentialgleichungen, ISNM 31, 147–163, Birkhäuser, Basel 1976
Meyer-Spasche, R.: A note on the approximation of mildly nonlinear Dirichlet problems by finite differences, Numer. Math. 33 (1979), 303–313
Mooney, J. W., Voss, H., Werner, B.: The dependence of critical parameter bounds on the monotonicity of a Newton sequence, Numer. Math. 33 (1979), 291–30l
Moore, G., Spence, A.: The convergence of Newton’s method near a bifurcation point, Technical Report Math/Na/4, University of Bath, 1977
Moore, G., Spence, A.: The calculation of turning points of nonlinear equations, Technical Report Math/Na/5, University of Bath, 1979, to appear in SIAM J. Numer. Math,
Peitgen, H. O., Walther, H. O. (eds.): Functional differential equations and approximation of fixed points, Proceedings Bonn 1978, Lecture Notes in Maths. 730, Springer-Verlag, 1979
Peitgen. H. O., Prüfer, M.: The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems, pp. 326–409 in |5l|
Pimbley, G. H. Jr.: Eigenfunction branches of nonlinear operators, and their bifurcations, Lecture Notes in Maths. 104, Springer-Verlag, 1969
Pönisch, G., Schwetlick, H.: Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven, Preprint, TU Dresden, 1977
Prüfer, M.: Calculating global bifurcation, pp. 187–214, in |85|
Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513
Rabinowitz, P. H. (ed.): Applications of bifurcation theory, Academic Press, New York 1977
Reddien, G. W.: On Newton’s method for singular problems, SIAM J. Numer. Anal. 15 (1978), 993–996
Reddien, G. W.: Newton’s method and higher order singularities, Computer Math. Appl. 5 (1979), 79–86
Reiss, E., L.: Imperfect bifurcation, pp. 37–72, in |57|
Rheinboldt, W. C.: Numerical methods for a class of finite dimensional bifurcation problems, SIAM J. Numer. Anal. 15 (1978), 1–11
Rheinboldt, W. C.: Numerical continuation methods for finite element applications, in: Bathe, K.-J., Oden, J. T., Wunderlich, W. (eds.): Formulations and computational algorithms in finite element analysis, MIT Press, 1977
Sattinger, D. H.: Topics in stability and bifurcation theory, Lecture Notes in Maths. 309, Springer-Verlag, 1973
Sattinger, D. H.: Group theoretic methods in bifurcation theory, Lecture Notes in Maths. 762, Springer-Verlag, 1979
Sattinger, D. H.: Recent progress in bifurcation theory, in: Cesari, L., Kannan, R., Weinberger, H. F. (eds,): Nonlinear Analysis, Academic Press, New York 1978
Scheurle, J.: Selective iteration and applications, J. Math. Anal. Appls. 59 (1977), 596–616
Scheurle, J.: Ein selektives Projektions-Iterationsverfahren und Anwendungen auf Verzweigungsprobleme, Numer. Math. 29 (1977), 11–35
Scholz, R.: A posteriori-Abschätzungen für Lösungen der stationären Navier -Stokes-Gleichungen bei Galerkin-Verfahren mit gemischten finiten Elementen, Habilitationsschrift, Universität Freiburg (1978)
Schröder, J.: Störungsrechnung bei Eigenwert- und Verzweigungsaufgaben, Arch. Rational Mech. Anal. 1 (1957), 436–468
Schwartz, J. T.: Compact analytical mapping of B-spaces and a theorem of Jane Cronin, Comm. Pure Appl. Math. 16 (1963), 253–260
Sermange, M.: Une méthode numérique en bifurcation-application à un probléme à frontière libre de la physique des plasmas, Appl. Math. Optim. 5 (1979), 127–151
Seydel, R.: Sumerische Berechnung von Verzweigungen bei gewöhnlichen Differentialgleichungen, Dissertation, TUM-MATH-7736, TU München (1977)
Seydel, R.: Numerical computation of branch points in ordinary differential equations, Numer. Math. 32 (1979), 51–68
Seydel, R.: Numerical computation of primary bifurcation points in ordinary differential equations, in: Albrecht, J., Collatz, L., Kirchgässner, K.: Constructive methods for nonlinear boundary value problems and nonlinear oscillations, ISNM 48, 161–169 Birkhäuser, Basel 1979
Seydel, R.: Numerical computation of branch points in nonlinear equations, Numer. Math. 33 (1979), 339–352
Simpson, R. B.: A method for the numerical determination of bifurcation states of nonlinear systems of equations, SIAM J. Numer. Anal. 12 (1975), 439–451
Sprekels, J.: Exact bounds for the solution branches of nonlinear eigenvalues problems, Numer. Math. 34 (1980), 29–40
Stakgold, I.: Branching of solutions of nonlinear equations, SIAM Review 13 (1971), 289–332
Trenogin, V. A., Sidorov, N. A.: The Tikhonov regularization of a problem on bifurcation points of nonlinear operators, Sibir. Math. J. 17 (1976), 314–323, translation from Sib. Mat. Zhurn. 17 (1976), 402–413
Vainberg, M. M.: Variational methods for the study of nonlinear operators, Holden Day Inc., San Francisco 1964
Vainberg, M. M., Trenogin, V. A., Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen, Akademie Verlag, Berlin 1973
Vainberg, M. M., Trenogin, V. A.: The methods of Lyapunov and Schmidt in the theory of non-linear equations and their further development, Russian Math. Surveys 17 (1962), 1–60
Voss, H., Werner, B.: Ein Quotienteneinschließungssatz für den kritischen Parameter nichtlinearer Randwertaufgaben in: Ansorge, R., Glashoff, K., Werner, B. (eds.): Numerische Mathematik, ISNM 49, 147–158, Birkhäuser, Basel 1979
Voss, H., Werner, B.: Bounds for the critical parameter of nonlinear eigenvalue problems, in preparation (1979)
Wacker, H. (ed.): Continuation methods, Academic Press, New York 1978
Wacker, H.: A summary of the developments of imbedding methods, pp. 1–35 in |85|
Weber, H.: Numerische Behandlung von Verzweigungsproblemen bei Randwertaufgaben gewöhnlicher Differentialgleichungen. Dissertation, Mainz 1978
Weber, H.: Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen. Numer. Math. 32 (1979) 17–29
Weber, H.: Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Randwertaufgaben, in: Albrecht, J., Collatz, L., Kirchgässner, K. (eds.): Constructive methods for nonlinear boundary value problems and nonlinear oscillations, ISNM 48, 176–190, Birkhäuser, Basel 1979
Weber, H.: Numerische Behandlung von Verzweigungsproblemen, in: Gorenflo, R. (ed.): Inkorrekt gestellte Probleme II, Preprint No. 79/78, FU Berlin
Weber, H.: Numerical solution of Hopf bifurcation problems, Math. Meth. Appl. Sci. 2 (1980), 178–190
Weber, H.: Eine Verallgemeinerung eines Satzes von Crandall und Rabinowitz aus der Verzweigungstheorie, Preprint Nr. 31 (1979), Universität Dortmund, Angewandte Mathematik, submitted for publication
Weber, H.: Shooting methods for bifurcation problems in ordinary differential equations, these proceedings
Weber, H., Werner W.: On the accurate determination of nonisolated solutions of nonlinear equations, Preprint Nr. 32 (1979), Universität Dortmund, Angewandte Mathematik, submitted for publication
Weiss, R.: Bifurcation in difference approximations to two-point boundary value problems, Math. Comp. 29 (1975), 746–760
Westreich, D., Vaarol, Y.L.: Numerical bifurcation at simple eigenvalues, SIAM J. Numer. Anal. 16 (1979), 538–546
Westreich, D., Vaarol, Y.L.: Applications of Galerkin’s method to bifurcation and two-point boundary value problems, J. Math. Anal. Appls. 70 (1979), 399–422
Wiesweg, U.: Verzweigungen bei Eigenwertaufgaben gewöhnlicher Differentialgleichungen und ihre numerische Behandlung, Diplomarbeit, Dortmund 1979
Yamaguti, M., Fujii, H.: On numerical deformation of singularities in nonlinear elasticity, in: Glowinski, R., Lions, J. L. (eds.): Computing methods in applied sciences and engineering, Springer Lecture Notes in Maths. 704 (1979)
Zeidler, E.: Some recent results in bifurcation theory, pp.185–202 in: Maurin, R., Raczka, R. (eds.): Mathematical physics and physical mathematics, Proc. Int. Symp. Warsaw 1974, Reidel, Dordrecht-Boston 1976.
Zeidler, E.: Vorlesungen über nichtlineare Funktionalanalysis I, Teubner, Leipzig 1976. Supplementary References:
Nekrassow, A. I.: Über Wellen vom permanenten Typ I/II (russ.), Polyt. Institut I. Wosnenski 3 (1921), 52–65 and 6 (1922), 155–171.
Hermann, M.: Zur integrativen Behandlung der Bifurkation von einfachen Eigenwerten bei gewöhnlichen Differentialgleichungen Forschungsergebnisse N/80/8, Universität Jena (1980).
Hassard, B.: A note on computation of periodic solutions of x =-αx(t-l) [1+x(t)], Preprint 1977.
Küpper, T.: Pointwise error bounds for the solutions of nonlinear boundary value problems, these proceedings
Moore, G., Spence, A.: The convergence of approximations to nonlinear equations at simple turning points, these proceedings
Moore, G.: The numerical treatment of non-trivial bifurcation points, Technical Report /Na/6, University of Bath (1980), submitted for publication
Seydel, R.: Programme zur numerischen Behandlung von Verzweigungsproblemen bei nichtlinearen Gleichungen und Differentialgleichungen, these proceedings
Kratochvil, A., Necas, J.: Gradient methods for the construction of Ljusternik-Schnirelmann critical values, R.A.I.R.O. Analyse numerique 14 (1980), 43–54
Scholz, R.: Computation of turning points of the stationary Navier-Stokes equations using mixed finite elements, these proceedings
Bohl, E.: Chord techniques and Newton’s method for discrete bifurcation problems, Numer. Math. 34 (1980), 111–124
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Mittelmann, H.D., Weber, H. (1980). Numerical Methods for Bifurcation Problems — A Survey and Classification. In: Mittelmann, H.D., Weber, H. (eds) Bifurcation Problems and their Numerical Solution. ISNM: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 54. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6294-3_1
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