Computation of Bifurcation Graphs

The numerical treatment of equivariant parameter-dependent nonlinear equation systems, and even more its automation requires the intensive use of group theory. This paper illustrates the group theoretic computations which are done in the preparation of the numerical computations. The bifurcation graph which gives the bifurcation subgroups is determined from the interrelationship of the irreducible representations of a group and its subgroups. The Jacobian is transformed to block diagonal structure using a modification of the transformation which transforms to block diagonal structure with respect to a supergroup. The principle of conjugacy is used everywhere to make symbolic and numerical computations even more efficient. Finally, when the symmetry reduced problems and blocks of Jacobian matrices are evaluated numerically, the fact that the given representation is a quasi-permutation representation is exploited automatically.

[1]  Vladimír Janovský,et al.  Numerical applications of equivariant reduction techniques , 1992 .

[2]  B. Werner,et al.  Eigenvalue Problems with the Symmetry of a Group and Bifurcations , 1990 .

[3]  K. Georg,et al.  Exploiting Symmetry in Solving Linear Equations , 1992 .

[4]  T. Healey.  A group-theoretic approach to computational bifurcation problems with symmetry , 1988 .

[5]  P. Deuflhard,et al.  Efficient numerical path following beyond critical points , 1987 .

[6]  Walter Knödel,et al.  Graphentheoretische Methoden und ihre Anwendungen , 1969 .

[7]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[8]  Kazuo Murota,et al.  Bifurcation analysis of symmetric structures using block-diagonalization , 1991 .

[9]  C. Curtis,et al.  Representation theory of finite groups and associated algebras , 1962 .

[10]  M. Dellnitz,et al.  Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points , 1989 .

[11]  André Vanderbauwhede,et al.  Local bifurcation and symmetry , 1982 .

[12]  B. Werner.  Test Functions for Bifurcation Points and Hopf Points in Problems with Symmetries , 1992 .

[13]  E. Stiefel,et al.  Gruppentheoretische Methoden und ihre Anwendung : eine Einführung mit typischen Beispielen aus Natur- und Ingenieurwissenschaften , 1979 .

[14]  Kazuo Murota,et al.  Computational Use of Group Theory in Bifurcation Analysis of Symmetric Structures , 1991, SIAM J. Sci. Comput..

[15]  Symmetry Adapted Block Diagonalisation in Equivariant Steady State Bifurcation Problems and its Numerical Application , 1991 .

[16]  Andreas Hohmann,et al.  Symbolic exploitation of symmetry in numerical pathfollowing , 1991, IMPACT Comput. Sci. Eng..

[17]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[18]  G. B. Robinson,et al.  Representation theory of the symmetric group , 1961 .

[19]  Ikeda Kiyohiro,et al.  Bifurcation hierarchy of symmetric structures , 1991 .

[20]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.