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Numerical Methods for Bifurcation Problems — A Survey and Classification

  • Hans Detlef Mittelmann
  • Helmut Weber
Chapter
Part of the ISNM: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 54)

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

  1. 1.
    Abbott, J. P.: An efficient algorithm for the determination of certain bifurcation points, J. Comp. Appl. Maths. 4 (1978), 19–27CrossRefGoogle Scholar
  2. 2.
    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 ReviewGoogle Scholar
  3. 3.
    Amann, H.: Ljusternik-Schnirelman theory and nonlinear eigenvalue problems, Math. Ann. 199 (1972), 55–72CrossRefGoogle Scholar
  4. 4.
    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), 476–501CrossRefGoogle Scholar
  5. 5.
    Atkinson, K. E.: The numerical solution of a bifurcation problem, SIAM J. Numer. Anal. 14 (1977), 584–599CrossRefGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Bazley, N. W.: Approximation of operators with reproducing non-linearities, manuscripta math. 18 (1976), 353–369CrossRefGoogle Scholar
  8. 8.
    Beyn, W.-J.: On discretizations of bifurcation problems, these proceedingsGoogle Scholar
  9. 9.
    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, 1978CrossRefGoogle Scholar
  10. 10.
    Bohl, E.: On the bifurcation diagram of discrete analogues of ordinary bifurcation problems, Math. Meth. Appl. Sci. 1 (1979), 566–571CrossRefGoogle Scholar
  11. 11.
    Bongers, A.: Behandlung verallgemeinerter nichtlinearer Eigenwertprobleme mit Ljusternik-Schnirelmann-Theorie, Dissertation, Mainz 1979Google Scholar
  12. 12.
    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), 211–224CrossRefGoogle Scholar
  13. 13.
    Crandall, M. G.: An introduction to constructive aspects of bifurcation and the implicit function theorem, in |57|Google Scholar
  14. 14.
    Crandall, M. G., Rabinowitz, P. H.: Bifurcation from simple eigenvalues, J. Functional Anal. 8 (1971), 321–340CrossRefGoogle Scholar
  15. 15.
    Crandall, M. G., Rabinowitz, P. H.: Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180CrossRefGoogle Scholar
  16. 16.
    Decker, D. W.: Some topics in bifurcation theory, PhD. thesis, Caltech, Pasadena, 1978Google Scholar
  17. 17.
    Demoulin, Y.-M. J., Chen, Y. M.: An iteration method of solving nonlinear eigenvalue problems, SIAM J. Numer. Anal. 28 (1975), 588–595Google Scholar
  18. 18.
    Drexler, F.-J.: Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen, Numer. Math. 29 (1977), 45–58CrossRefGoogle Scholar
  19. 19.
    Drexler, F.-J.: A homotopy-method for the calculation of all zeros of zero-dimensional polynomial ideals, pp. 64–93 in |85|Google Scholar
  20. 20.
    Engl, H.: On the change of parameters in continuation methods, Preprint Univ. Linz, Math. Inst. Nr. 58 (1976)Google Scholar
  21. 21.
    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 1979Google Scholar
  22. 22.
    Georg, K.: On the convergence of an inverse iteration method for nonlinear elliptic eigenvalue problems, Numer. Math. 32 (1979)Google Scholar
  23. 23.
    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 JenaGoogle Scholar
  24. 24.
    Hoffmann W., Küpper, T.: Punktweise Abschätzungen zur Ermittlung des Einflusses von Störtermen bei Verzweigungsproblemen, Report 78–22, Math. Institut der Universität zu Köln (1978)Google Scholar
  25. 25.
    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|Google Scholar
  26. 26.
    Iudovich, V. I.: The onset of auto-oscillations in a fluid, J. Appl. Math. Mech. 35(1971), 587–603CrossRefGoogle Scholar
  27. 27.
    Jeggle, H.: Existence and discrete approximation of bifurcation points, pp. 157–186 in |85|Google Scholar
  28. 28.
    Keener, J. P., Keller, H. B.: Perturbed bifurcation theory, Arch. Rational Mech. Anal. 50(1974), 159–175CrossRefGoogle Scholar
  29. 29.
    Keller, H. B.: Numerical solution of bifurcation and nonlinear eigenvalue problems, pp. 359–384 in |57|Google Scholar
  30. 30.
    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, 1979Google Scholar
  31. 31.
    Keller, H. B., Langford, W. F.: Iterations, perturbations and multiplicities for nonlinear bifurcation problems, Arch. Rational Mech. Anal. 48 (1972), 83–108Google Scholar
  32. 32.
    Keller, J. B., Antman, S. (eds.): Bifurcation theory and nonlinear eigenvalue problems, Benjamin, New York 1969Google Scholar
  33. 33.
    Kesavan, S.: La méthode de Kikuchi appliquée aux équations de von Karman, Numer. Math. 32 (1979), 209–232CrossRefGoogle Scholar
  34. 34.
    Kikuchi, F.: An iterative finite element scheme for bifurcation analysis of semi-linear elliptic equations, ISAS Report No, 542, Tokyo 1976Google Scholar
  35. 35.
    Kikuchi, F.: Finite element approximation of bifurcation problems, Theoretical and Applied Mechanics 26 (1976), 37–51, University of Tokyo PressGoogle Scholar
  36. 36.
    Kikuchi, F.: Numerical analysis of the finite element method applied to bifurcation problems of turning point type, ISAS Report No. 564, Tokyo 1978Google Scholar
  37. 37.
    Kikuchi, F.: Finite element approximations to bifurcation problems of turning point type, Theoretical and Applied Mechanics 27 (1977), 99–114, University of Tokyo PressGoogle Scholar
  38. 38.
    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, 1979Google Scholar
  39. 39.
    Krasnosel’skii, M. A.: Topological methods in the theory of nonlinear integral equations, Pergamon Press, Oxford 1964Google Scholar
  40. 40.
    Kubicek, M.: Evaluation of branching-points for nonlinear boundary-value problems based on the GPM technique, Appl. Maths. Comp. 1(1975), 341–352CrossRefGoogle Scholar
  41. 41.
    Kubicek, M., Marek, M.: Evaluation of limit and bifurcation points for algebraic equations and nonlinear boundary value problems, Appl. Maths. Comp. 5 (1979), 253–264CrossRefGoogle Scholar
  42. 42.
    Langford, W. F.: A shooting algorithm for the best least squares solution of two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977), 527–542CrossRefGoogle Scholar
  43. 43.
    Langford, W. F.: Numerical solution of bifurcation problems for ordinary differential equations, Numer. Math. 28 (1977), 171–190CrossRefGoogle Scholar
  44. 44.
    Leray, J., Schauder, J. P.: Topologie et equations fonctionelles, Ann. Ecole Norm. Sup. (3) 51 (1934), 45–78Google Scholar
  45. 45.
    Marsden, J. E., McCracken, M.: The Hopf bifurcation and its applications, Springer-Verlag, New York 1976CrossRefGoogle Scholar
  46. 46.
    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 1976CrossRefGoogle Scholar
  47. 47.
    Meyer-Spasche, R.: A note on the approximation of mildly nonlinear Dirichlet problems by finite differences, Numer. Math. 33 (1979), 303–313CrossRefGoogle Scholar
  48. 48.
    Mooney, J. W., Voss, H., Werner, B.: The dependence of critical parameter bounds on the monotonicity of a Newton sequence, Numer. Math. 33 (1979), 291–30lCrossRefGoogle Scholar
  49. 49.
    Moore, G., Spence, A.: The convergence of Newton’s method near a bifurcation point, Technical Report Math/Na/4, University of Bath, 1977Google Scholar
  50. 50.
    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,Google Scholar
  51. 51.
    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, 1979Google Scholar
  52. 52.
    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|Google Scholar
  53. 53.
    Pimbley, G. H. Jr.: Eigenfunction branches of nonlinear operators, and their bifurcations, Lecture Notes in Maths. 104, Springer-Verlag, 1969CrossRefGoogle Scholar
  54. 54.
    Pönisch, G., Schwetlick, H.: Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven, Preprint, TU Dresden, 1977Google Scholar
  55. 55.
    Prüfer, M.: Calculating global bifurcation, pp. 187–214, in |85|Google Scholar
  56. 56.
    Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513CrossRefGoogle Scholar
  57. 57.
    Rabinowitz, P. H. (ed.): Applications of bifurcation theory, Academic Press, New York 1977Google Scholar
  58. 58.
    Reddien, G. W.: On Newton’s method for singular problems, SIAM J. Numer. Anal. 15 (1978), 993–996CrossRefGoogle Scholar
  59. 59.
    Reddien, G. W.: Newton’s method and higher order singularities, Computer Math. Appl. 5 (1979), 79–86CrossRefGoogle Scholar
  60. 60.
    Reiss, E., L.: Imperfect bifurcation, pp. 37–72, in |57|Google Scholar
  61. 61.
    Rheinboldt, W. C.: Numerical methods for a class of finite dimensional bifurcation problems, SIAM J. Numer. Anal. 15 (1978), 1–11CrossRefGoogle Scholar
  62. 62.
    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, 1977Google Scholar
  63. 63.
    Sattinger, D. H.: Topics in stability and bifurcation theory, Lecture Notes in Maths. 309, Springer-Verlag, 1973Google Scholar
  64. 64.
    Sattinger, D. H.: Group theoretic methods in bifurcation theory, Lecture Notes in Maths. 762, Springer-Verlag, 1979Google Scholar
  65. 65.
    Sattinger, D. H.: Recent progress in bifurcation theory, in: Cesari, L., Kannan, R., Weinberger, H. F. (eds,): Nonlinear Analysis, Academic Press, New York 1978Google Scholar
  66. 66.
    Scheurle, J.: Selective iteration and applications, J. Math. Anal. Appls. 59 (1977), 596–616CrossRefGoogle Scholar
  67. 67.
    Scheurle, J.: Ein selektives Projektions-Iterationsverfahren und Anwendungen auf Verzweigungsprobleme, Numer. Math. 29 (1977), 11–35CrossRefGoogle Scholar
  68. 68.
    Scholz, R.: A posteriori-Abschätzungen für Lösungen der stationären Navier -Stokes-Gleichungen bei Galerkin-Verfahren mit gemischten finiten Elementen, Habilitationsschrift, Universität Freiburg (1978)Google Scholar
  69. 69.
    Schröder, J.: Störungsrechnung bei Eigenwert- und Verzweigungsaufgaben, Arch. Rational Mech. Anal. 1 (1957), 436–468CrossRefGoogle Scholar
  70. 70.
    Schwartz, J. T.: Compact analytical mapping of B-spaces and a theorem of Jane Cronin, Comm. Pure Appl. Math. 16 (1963), 253–260CrossRefGoogle Scholar
  71. 71.
    Sermange, M.: Une méthode numérique en bifurcation-application à un probléme à frontière libre de la physique des plasmas, Appl. Math. Optim. 5 (1979), 127–151CrossRefGoogle Scholar
  72. 72.
    Seydel, R.: Sumerische Berechnung von Verzweigungen bei gewöhnlichen Differentialgleichungen, Dissertation, TUM-MATH-7736, TU München (1977)Google Scholar
  73. 73.
    Seydel, R.: Numerical computation of branch points in ordinary differential equations, Numer. Math. 32 (1979), 51–68CrossRefGoogle Scholar
  74. 74.
    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 1979Google Scholar
  75. 75.
    Seydel, R.: Numerical computation of branch points in nonlinear equations, Numer. Math. 33 (1979), 339–352CrossRefGoogle Scholar
  76. 76.
    Simpson, R. B.: A method for the numerical determination of bifurcation states of nonlinear systems of equations, SIAM J. Numer. Anal. 12 (1975), 439–451CrossRefGoogle Scholar
  77. 77.
    Sprekels, J.: Exact bounds for the solution branches of nonlinear eigenvalues problems, Numer. Math. 34 (1980), 29–40CrossRefGoogle Scholar
  78. 78.
    Stakgold, I.: Branching of solutions of nonlinear equations, SIAM Review 13 (1971), 289–332CrossRefGoogle Scholar
  79. 79.
    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), 402–413CrossRefGoogle Scholar
  80. 80.
    Vainberg, M. M.: Variational methods for the study of nonlinear operators, Holden Day Inc., San Francisco 1964Google Scholar
  81. 81.
    Vainberg, M. M., Trenogin, V. A., Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen, Akademie Verlag, Berlin 1973Google Scholar
  82. 82.
    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), 1–60CrossRefGoogle Scholar
  83. 83.
    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 1979Google Scholar
  84. 84.
    Voss, H., Werner, B.: Bounds for the critical parameter of nonlinear eigenvalue problems, in preparation (1979)Google Scholar
  85. 85.
    Wacker, H. (ed.): Continuation methods, Academic Press, New York 1978Google Scholar
  86. 86.
    Wacker, H.: A summary of the developments of imbedding methods, pp. 1–35 in |85|Google Scholar
  87. 87.
    Weber, H.: Numerische Behandlung von Verzweigungsproblemen bei Randwertaufgaben gewöhnlicher Differentialgleichungen. Dissertation, Mainz 1978Google Scholar
  88. 88.
    Weber, H.: Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen. Numer. Math. 32 (1979) 17–29CrossRefGoogle Scholar
  89. 89.
    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 1979Google Scholar
  90. 90.
    Weber, H.: Numerische Behandlung von Verzweigungsproblemen, in: Gorenflo, R. (ed.): Inkorrekt gestellte Probleme II, Preprint No. 79/78, FU BerlinGoogle Scholar
  91. 91.
    Weber, H.: Numerical solution of Hopf bifurcation problems, Math. Meth. Appl. Sci. 2 (1980), 178–190CrossRefGoogle Scholar
  92. 92.
    Weber, H.: Eine Verallgemeinerung eines Satzes von Crandall und Rabinowitz aus der Verzweigungstheorie, Preprint Nr. 31 (1979), Universität Dortmund, Angewandte Mathematik, submitted for publicationGoogle Scholar
  93. 93.
    Weber, H.: Shooting methods for bifurcation problems in ordinary differential equations, these proceedingsGoogle Scholar
  94. 94.
    Weber, H., Werner W.: On the accurate determination of nonisolated solutions of nonlinear equations, Preprint Nr. 32 (1979), Universität Dortmund, Angewandte Mathematik, submitted for publicationGoogle Scholar
  95. 95.
    Weiss, R.: Bifurcation in difference approximations to two-point boundary value problems, Math. Comp. 29 (1975), 746–760CrossRefGoogle Scholar
  96. 96.
    Westreich, D., Vaarol, Y.L.: Numerical bifurcation at simple eigenvalues, SIAM J. Numer. Anal. 16 (1979), 538–546CrossRefGoogle Scholar
  97. 97.
    Westreich, D., Vaarol, Y.L.: Applications of Galerkin’s method to bifurcation and two-point boundary value problems, J. Math. Anal. Appls. 70 (1979), 399–422CrossRefGoogle Scholar
  98. 98.
    Wiesweg, U.: Verzweigungen bei Eigenwertaufgaben gewöhnlicher Differentialgleichungen und ihre numerische Behandlung, Diplomarbeit, Dortmund 1979Google Scholar
  99. 99.
    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)Google Scholar
  100. 100.
    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.Google Scholar
  101. 101.
    Zeidler, E.: Vorlesungen über nichtlineare Funktionalanalysis I, Teubner, Leipzig 1976. Supplementary References:Google Scholar
  102. 102.
    Nekrassow, A. I.: Über Wellen vom permanenten Typ I/II (russ.), Polyt. Institut I. Wosnenski 3 (1921), 52–65 and 6 (1922), 155–171.Google Scholar
  103. 103.
    Hermann, M.: Zur integrativen Behandlung der Bifurkation von einfachen Eigenwerten bei gewöhnlichen Differentialgleichungen Forschungsergebnisse N/80/8, Universität Jena (1980).Google Scholar
  104. 104.
    Hassard, B.: A note on computation of periodic solutions of x =-αx(t-l) [1+x(t)], Preprint 1977.Google Scholar
  105. 105.
    Küpper, T.: Pointwise error bounds for the solutions of nonlinear boundary value problems, these proceedingsGoogle Scholar
  106. 106.
    Moore, G., Spence, A.: The convergence of approximations to nonlinear equations at simple turning points, these proceedingsGoogle Scholar
  107. 107.
    Moore, G.: The numerical treatment of non-trivial bifurcation points, Technical Report /Na/6, University of Bath (1980), submitted for publicationGoogle Scholar
  108. 108.
    Seydel, R.: Programme zur numerischen Behandlung von Verzweigungsproblemen bei nichtlinearen Gleichungen und Differentialgleichungen, these proceedingsGoogle Scholar
  109. 109.
    Kratochvil, A., Necas, J.: Gradient methods for the construction of Ljusternik-Schnirelmann critical values, R.A.I.R.O. Analyse numerique 14 (1980), 43–54Google Scholar
  110. 110.
    Scholz, R.: Computation of turning points of the stationary Navier-Stokes equations using mixed finite elements, these proceedingsGoogle Scholar
  111. 111.
    Bohl, E.: Chord techniques and Newton’s method for discrete bifurcation problems, Numer. Math. 34 (1980), 111–124CrossRefGoogle Scholar

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© Springer Basel AG 1980

Authors and Affiliations

  • Hans Detlef Mittelmann
    • 1
  • Helmut Weber
    • 1
  1. 1.Abteilung MathmatikUniversität DortmurdGermany

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