Numerical Methods for Bifurcation Problems — A Survey and Classification

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.

[1]  G. Moore,et al.  The Calculation of Turning Points of Nonlinear Equations , 1980 .

[2]  J. Leray,et al.  Topologie et équations fonctionnelles , 1934 .

[3]  J. Schwartz Compact analytic mappings of B-spaces and a theorem of Jane Cronin , 1963 .

[4]  M. Kubíek Evaluation of branching points for nonlinear boundary-value problems based on the GPM technique , 1975 .

[5]  Approximation of operators with reproducing nonlinearities , 1976 .

[6]  On a problem of forced nonlinear oscillations. numerical example of bifurcation into an invariant torus , 1978 .

[7]  J. Abbott An e cient algorithm for the determination of certain bifurcation points , 1978 .

[8]  M. Yamaguti,et al.  On numerical deformation of singularities in nonlinear elasticity , 1979 .

[9]  Numerical Bifurcation at Simple Eigenvalues , 1979 .

[10]  R. Meyer-Spasche A note on the approximation of mildly nonlinear Dirichlet problems by finite differences , 1979 .

[11]  An Iterative Finite Element Scheme for Bifurcation Analysis of Semi-Linear Elliptic Equations , 1976 .

[12]  Rüdiger Seydel Programme zur Numerischen Behandlung von Verzweigungsproblemen bei Nichtlinearen Gleichungen und Differentialgleichungen , 1980 .

[13]  Kurt Georg An Iteration Method for Solving Nonlinear Eigenvalue Problems , 1979 .

[14]  R. B. Simpson A Method for the Numerical Determination of Bifurcation States of Nonlinear Systems of Equations , 1975 .

[15]  S. S. Antman,et al.  Bifurcation theory and nonlinear eigenvalue problems, 1967 , 1968 .

[16]  Jürgen Scheurle Ein selektives Projektions-Iterationsverfahren und Anwendungen auf Verzweigungsprobleme , 1977 .

[17]  Evaluation of limit and bifurcation points for algebraic equations and nonlinear boundary-value problems , 1979 .

[18]  Lothar Collatz,et al.  Numerische Behandlung von Differentialgleichungen , 1948 .

[19]  W. Langford Numerical solution of bifurcation problems for ordinary differential equations , 1977 .

[20]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[21]  G. H. Pimbley On eigenfunction branches of nonlinear operators, and their bifurcations , 1969 .

[22]  G. Reddien On Newton’s Method for Singular Problems , 1978 .

[23]  D. Westreich,et al.  Applications of Galerkin's method to bifurcation and two-point boundary value problems☆ , 1979 .

[24]  The Tikhonov regularization of a problem on bifurcation points of nonlinear operators , 1976 .

[25]  Michael G. Crandall,et al.  Bifurcation, perturbation of simple eigenvalues, itand linearized stability , 1973 .

[26]  R. Seydel Numerical computation of branch points in ordinary differential equations , 1979 .

[27]  H. Keller,et al.  Iterations, perturbations and multiplicities for nonlinear bifurcation problems , 1972 .

[28]  M. Sermange,et al.  Une méthode numérique en bifurcation—application à un problème à frontière libre de la physique des plasmas , 1979 .

[29]  Kurt Georg,et al.  On the convergence of an inverse iteration method for nonlinear elliptic eigenvalue problems , 1979 .

[30]  H. Keller,et al.  Perturbed bifurcation theory , 1973 .

[31]  H. Weber,et al.  Numerical solution of Hopf bifurcation problems , 1980 .

[32]  Klaus Kirchgässner,et al.  Constructive methods for nonlinear boundary value problems and nonlinear oscillations : conference at the Oberwolfach Mathematical Research Institute, Black Forest, November 19-25, 1978 , 1979 .

[33]  P. Anselone,et al.  AN EXTENSION OF THE NEWTON-KANTOROVIC METHOD FOR SOLVING NONLINEAR EQUATIONS, , 1966 .

[34]  D. Decker,et al.  Topics in bifurcation theory , 1978 .

[35]  R. Meyer-Spasche Numerical Treatment of Dirichlet Problems with Several Solutions , 1976 .

[36]  E. Bohl On two boundary value problems in nonlinear elasticity from a numerical viewpoint , 1978 .

[37]  E. Bohl,et al.  On the bifurcation diagram of discrete analogues for ordinary bifurcation problems , 1979 .

[38]  H. Peitgen,et al.  Functional Differential Equations and Approximation of Fixed Points , 1979 .

[39]  M. Crandall,et al.  Bifurcation from simple eigenvalues , 1971 .

[40]  M. M. Vaĭnberg,et al.  THE METHODS OF LYAPUNOV AND SCHMIDT IN THE THEORY OF NON-LINEAR EQUATIONS AND THEIR FURTHER DEVELOPMENT , 1962 .

[41]  G. Reddien Newton's method and high order singularities , 1979 .

[42]  Willi Törnig,et al.  Numerical Treatment of Differential Equations in Applications , 1978 .

[43]  F. Drexler Eine Methode zur berechnung sämtlicher Lösungen von Polynomgleichungssystemen , 1977 .

[44]  H. Amann Lusternik-Schnirelman theory and non-linear eigenvalue problems , 1972 .

[45]  E. Bohl Chord techniques and Newton's method for discrete bifurcation problems , 1980 .

[46]  A. Feinstein,et al.  Variational Methods for the Study of Nonlinear Operators , 1966 .

[47]  H. B. Keller,et al.  Constructive methods for bifurcation and nonlinear eigenvalue problems , 1979 .

[48]  Paul H. Rabinowitz,et al.  Some global results for nonlinear eigenvalue problems , 1971 .

[49]  G. Moore,et al.  The numerical treatment of non-trivial bifurcation points , 1980 .

[50]  W. Langford A Shooting Algorithm for the Best Least Squares Solution of Two-Point Boundary Value Problems , 1977 .

[51]  H. Weber Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen , 1979 .

[52]  R. Seydel Numerical computation of branch points in nonlinear equations , 1979 .

[53]  Alexander Kratochvíl,et al.  Gradient methods for the construction of Ljusternik-Schnirelmann critical values , 1980 .

[54]  Wainberg, M. M./Trenogin, W. A., Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen. (Math. Lehrbücher u. Monographien, B. 32). 408 S. m. 19 Abb. Berlin 1973. Akademie-Verlag. Preis gob. 54,— M , 1975 .

[55]  La méthode de Kikuchi appliquée aux équations de von Karman , 1979 .

[56]  M. Prüfer,et al.  The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems , 1979 .

[57]  R. Seydel Numerical Computation of Primary Bifurcation Points in Ordinary Differential Equations , 1979 .

[58]  H. Weber Shooting Methods for Bifurcation Problems in Ordinary Differential Equations , 1980 .

[59]  H. Weber Numerische Behandlung von Verzweigungsproblemen bei Gewöhnlichen Randwertaufgaben , 1979 .

[60]  The Numerical Solution of a Bifurcation Problem , 1977 .

[61]  J. Schröder Störungsrechnung bei Eigenwert- und Verzweigungsaufgaben , 1957 .

[62]  W. Rheinboldt Numerical methods for a class of nite dimensional bifur-cation problems , 1978 .

[63]  E. Allgower,et al.  Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations , 1980 .

[64]  Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen , 1973 .

[65]  D. Sattinger Topics in stability and bifurcation theory , 1973 .

[66]  F. Kikuchi Finite element approximations to bifurcation problems of turning point type , 1979 .

[67]  David H. Sattinger,et al.  Group theoretic methods in bifurcation theory , 1979 .

[68]  R. Weiss Bifurcation in difference approximations to two-point boundary value problems , 1975 .

[69]  H. Voss,et al.  The dependence of critical parameter bounds on the monotonicity of a Newton sequence , 1979 .

[70]  Y. M. Chen,et al.  An Iteration Method for Solving Nonlinear Eigenvalue Problems , 1975 .

[71]  M. A. Krasnoselʹskii Topological methods in the theory of nonlinear integral equations , 1968 .

[72]  J. Sprekels Exact bounds for the solution branches of nonlinear eigenvalue problems , 1980 .

[73]  V. I. Iudovich The onset of auto-oscillations in a fluid , 1971 .

[74]  J. Scheurle Selective iteration and applications , 1977 .

[75]  I. Stakgold,et al.  Branching of Solutions of Nonlinear Equations , 1971 .

[76]  R. Glowinski,et al.  Computing Methods in Applied Sciences and Engineering , 1974 .