Abstract
Bifurcation theory studies the behavior of multiple solutions of nonlinear (differential) equations when parameters in these equations are varied, and describes how the number and type of these solutions change. It is a domain of applied mathematics which uses concepts from such diverse fields as functional analysis, group representations, ideal theory and many others. For real, e.g. physically motivated problems, the calculations necessary to determine even the simplest bifurcations become excessively complicated. Therefore, a project to build a package "bifurcation and singularity theory" in computer algebra is presented. Specifically, Gröbner bases are used to determine the codimension of a singularity, thereby extending the Buchberger Algorithm to modules. Also, a program in SMP is described, which permits determining whether a given function g is contact equivalent to a polynomial normal form h for one dimensional bifurcation problems up to codimension three.
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References
D. Armbruster: Possible applications of computer algebra in singularity and bifurcation theory. Institute report, University of Tübingen, 1984.
B. Buchberger: Gröbner bases: An algorithmic method in polynomial ideal theory, Chapter 6 in: Recent Trends in Multidimensional Systems Theory (N.K. Bose, Ed.), Reidel 1985.
G. Dangelmayr, D. Armbruster: Proc. London Math. Soc. 46 (3), 517, (1983); D. Armbruster, G. Dangelmayr, W. Güttinger: Physica D (May 1985).
R. Gebauer, H. Kredel: Buchberger Algorithm System, Sigsam 18 (1), (1984).
M. Golubitsky, D. Schaeffer: Comm. Pure Appl. Math. 32, 21 (1979); M. Golubitsky, D. Schaeffer: Singularities and groups in bifurcation theory I, Springer 1985.
M. Golubitsky, W.F. Langford: J. Differential Equations 41 (3), 375 (1981).
M. Golubitsky, B.L. Keyfitz, D. Schaeffer: Comm. Pure Appl. Math. 34, 433 (1981).
J. Guckenheimer, Ph. Holmes: Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer 1983.
R.H. Rand, W.L. Keith: Normal form and center manifold calculations on MACSYMA. To appear in "Applications of Computer Algebra", R. Pavelle, Ed.
M.M. Vainberg, V.A. Trenogin: Theory of branching of solutions of nonlinear equations, Noordhoff 1974.
F.J. Wright, G. Dangelmayr: Reduction of univariate catastrophes to normal forms, preprint Tübingen 1984.
K. Millington, F.J. Wright: Algebraic computation in elementary catastrophe theory, these proceedings.
SMP Reference Manual, Inference Corporation 1983.
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© 1985 Springer-Verlag Berlin Heidelberg
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Armbruster, D. (1985). Bifurcation theory and computer algebra: An initial approach. In: Caviness, B.F. (eds) EUROCAL '85. EUROCAL 1985. Lecture Notes in Computer Science, vol 204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-15984-3_245
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DOI: https://doi.org/10.1007/3-540-15984-3_245
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