Abstract
Various quite satisfactory analytical and numerical techniques are available for analysing bifurcation points when something about the structure is known á priori. The author previously introduced a method applicable when such information is not present, or when the arcs intersect tangentially. That method is discussed here, with particular emphasis on avenues to improvement in efficiency and reliability.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. L. Allgower and K. Georg: Simplicial and continuation methods for approximating fixed points and solutions to systems of equations, SIAM Review 22 no. 1 (1980), pp. 28–85.
C.B. Garcia and W. I. Zangwill: Finding all solutions to polynomial systems and other systems of equations, Mathematical Programming 16 (1979), pp. 159–176.
C. B. Garcia ariiTW; I. Zangwill: Pathways to Solutions, Fixed Points, and Equilibria, Prentice-Hall, Englewood Cliffs, 1981.
C. B. Garcia and T.Y. Li: On the number of solutions to polynomial systems of equations, SIAM J. Numer. Anal. 17 no. 4 (1980), pp. 540–546.
M. Golubitsky and D. Schaeffer: A theory for imperfect bifurcations via singularity theory, Comm. Pure and Applied Math. 32 (1979), pp. 21–98.
M. Golubitsky and W. F. Langford: Classification and unfoldings of degenerate Hopf bifurcations, J. Diff. Eq. 41 no. 3 (1981), pp. 375–415.
R. B. Kearfott: An efficient degree computation method for a generalized method of bisection, Numer. Math. 32 (1979), pp. 109–127.
R. B. Kearfott: Some general bifurcation techniques, SIAM J. Sei. Stat. Comput. 4 no. 1 (1983), pp. 52–68.
R. B. Kearfott: An improved program for generalized bisection, to appear.
R. B. Kearfott: Analysis of a general bifurcation technique, to appear.
H. B. Keller: Numerical Solution of bifurcation and nonlinear eigenvalue problems, in Applications of Bifurcation Theory, ed. P.H. Rabinowitz., Academic Press, New York, 1977.
H.B. Keller and W. F. Langford: Iterations, perturbations, and multiplicities for nonlinear bifurcation problems, Arch. Rat. Mech. Anal. 48 (1972), pp. 83–108.
M. Kubicek: Dependence of Solution of nonlinear systems on a parameter, ACM TOMS 2 no. 1 (1976), pp. 98–107.
A. P. Morgan: A method for Computing all solutions to systems of polynomial equations, ACM TOMS 9 no. 1 (1983), pp. 1–17.
A. P. Morgan: Computing all solutions to polynomial systems using homogeneous coordinates in Euclidean space, in Numerical Analysis of Parametrized Nonlinear Equations, University of Arkansas seventh lecture series, 1983.
H. -O. Peitgen and M. Prüfer: The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems, in Functional Differential Equations and Approximation of Fixed Points, Springer Lecture Notes no. 730, New York, 1979.
L. B. Rall: Differentiation in PASCAL-SC: type gradient, Mathematics Research Center technical report no. 2400, University of Wisconsin, Madison, 1982.
W. C. Rheinboldt and J. V. Burkardt: A program for a locally-parametrized continuation process, ACM TOMS 9 no. 2 (1983), pp. 236–241.
W. C. Rheinboldt and J. V. Burkardt: A locally parametrized continuation process, ACM TOMS 9 no. 2 (1983), pp. 215–235.
L. T. Watson and D. Fenner: The Chow-Yorke algorithm for fixed points or zeros of C2 maps, ACM TOMS 6 (1980), pp. 252–260.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer Basel AG
About this chapter
Cite this chapter
Kearfott, R.B. (1984). On a General Technique for Finding Directions Proceeding from Bifurcation Points. In: Küpper, T., Mittelmann, H.D., Weber, H. (eds) Numerical Methods for Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 70. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6256-1_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6256-1_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-6257-8
Online ISBN: 978-3-0348-6256-1
eBook Packages: Springer Book Archive