Skip to main content

On a General Technique for Finding Directions Proceeding from Bifurcation Points

  • Chapter
Numerical Methods for Bifurcation Problems

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.

    Article  Google Scholar 

  2. 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.

    Article  Google Scholar 

  3. C. B. Garcia ariiTW; I. Zangwill: Pathways to Solutions, Fixed Points, and Equilibria, Prentice-Hall, Englewood Cliffs, 1981.

    Google Scholar 

  4. 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.

    Article  Google Scholar 

  5. M. Golubitsky and D. Schaeffer: A theory for imperfect bifurcations via singularity theory, Comm. Pure and Applied Math. 32 (1979), pp. 21–98.

    Article  Google Scholar 

  6. M. Golubitsky and W. F. Langford: Classification and unfoldings of degenerate Hopf bifurcations, J. Diff. Eq. 41 no. 3 (1981), pp. 375–415.

    Article  Google Scholar 

  7. R. B. Kearfott: An efficient degree computation method for a generalized method of bisection, Numer. Math. 32 (1979), pp. 109–127.

    Google Scholar 

  8. R. B. Kearfott: Some general bifurcation techniques, SIAM J. Sei. Stat. Comput. 4 no. 1 (1983), pp. 52–68.

    Article  Google Scholar 

  9. R. B. Kearfott: An improved program for generalized bisection, to appear.

    Google Scholar 

  10. R. B. Kearfott: Analysis of a general bifurcation technique, to appear.

    Google Scholar 

  11. 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.

    Google Scholar 

  12. H.B. Keller and W. F. Langford: Iterations, perturbations, and multiplicities for nonlinear bifurcation problems, Arch. Rat. Mech. Anal. 48 (1972), pp. 83–108.

    Article  Google Scholar 

  13. M. Kubicek: Dependence of Solution of nonlinear systems on a parameter, ACM TOMS 2 no. 1 (1976), pp. 98–107.

    Article  Google Scholar 

  14. A. P. Morgan: A method for Computing all solutions to systems of polynomial equations, ACM TOMS 9 no. 1 (1983), pp. 1–17.

    Article  Google Scholar 

  15. 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.

    Google Scholar 

  16. 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.

    Chapter  Google Scholar 

  17. L. B. Rall: Differentiation in PASCAL-SC: type gradient, Mathematics Research Center technical report no. 2400, University of Wisconsin, Madison, 1982.

    Google Scholar 

  18. W. C. Rheinboldt and J. V. Burkardt: A program for a locally-parametrized continuation process, ACM TOMS 9 no. 2 (1983), pp. 236–241.

    Article  Google Scholar 

  19. W. C. Rheinboldt and J. V. Burkardt: A locally parametrized continuation process, ACM TOMS 9 no. 2 (1983), pp. 215–235.

    Article  Google Scholar 

  20. 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.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints 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

Publish with us

Policies and ethics