Skip to main content
SpringerLink
Log in

Numerical computation of nonsimple turning points and cusps

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

A method is developed for the numerical computation of a double turning point corresponding to a cusp catastrophe of a nonlinear operator equation depending on two parameters. An augmented system containing the original equation is introduced, for which the cusp point is an isolated solution. An efficient implementation of Newton's method in the finite-dimensional case is presented. Results are given for some chemical engineering problems and this direct method is compared with some other techniques to locate cusp points.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beyn, W.J.: On discretizations of bifurcation problems. In [16], pp. 46–73

    Google Scholar 

  2. Caluwaerts, R.: A direct method for the determination of non-simple turning points. (To be published in Numer. Funct. Anal. Optimization)

  3. Chan, T.: Deflation techniques and block-elimination algorithms for solving bordered singular systems. SIAM J. Sci. Stat. Comput.5, 121–134 (1984)

    Google Scholar 

  4. Chan, T.: Newton-like pseudo-arclength methods for computing simple turning points. SIAM J. Sci. Stat. Comput.5, 135–148 (1984)

    Google Scholar 

  5. Brezzi, F., Rappaz, T., Raviart, P.A.: Finite dimensional approximation of nonlinear problems. II: Limit Points. Numer. Math.37, 1–28 (1981)

    Google Scholar 

  6. Hlavacek, V., Marek, M., Kubicek, M.: Modelling of chemical reactors-X. Multiple solutions of entalpy and mass balances for a catalytic reaction within a porous catalyst particle. Chem. Engng. Sci.23, 1083–1097 (1968)

    Google Scholar 

  7. Joyce, D.C.: Survey of extrapolation processes in Numerical Analysis. SIAM Rev.13, 435–488 (1971)

    Google Scholar 

  8. Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Applications of bifurcation theory. P.H. Rabinowitz (ed.), pp. 359–384. New York: Academic Press 1977

    Google Scholar 

  9. Keller, H.B.: Geometrically isolated nonisolated solutions and their approximation. SIAM J. Numer. Anal.18, 822–838 (1981)

    Google Scholar 

  10. Kordylewsky, W.: Degenerate critical points of thermal explosion. Kommunikat I-10/K-020/78. Technical University of Wroclaw 1978

  11. Kordylewsky, W.: Non-stationary method for calculating critical parameters of thermal explosion. In: Numerical Methods in Thermal Problems. R.W. Lewis, K. Morgan (eds.), pp. 1032–1041. Swansea: Pineridge Press 1979

    Google Scholar 

  12. Melhem, R.G., Rheinboldt, W.C.: A comparison of methods for determining turning points of nonlinear equations. Computing29, 201–226 (1982)

    Google Scholar 

  13. Mittelmann, H.D., Weber, H. (eds.): Bifurcation problems and their numerical solution. ISNM Vol. 54. Basel: Birkhäuser Verlag 1980

    Google Scholar 

  14. Moore, G., Spence, A.: The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal.17, 567–576 (1980)

    Google Scholar 

  15. Moore, G., Spence, A.: The convergence of operator equations at turning points. IMA J. Numer. Anal.1, 23–38 (1981)

    Google Scholar 

  16. Pönisch, G., Schwetlick, H.: Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter. Computing26, 107–121 (1981)

    Google Scholar 

  17. Poston, T., Stewart, I.N.: Catastrophe theory and its applications. London: Pitman Press 1978

    Google Scholar 

  18. Rheinboldt, W.C., Burkardt, J.V.: A locally parametrized continuation process. TOMS9, 215–241 (1983)

    Google Scholar 

  19. Roose, D., Piessens, R.: Numerical computation of nonsimple turning points and cusps. Report TW 60, Katholieke Universiteit Leuven 1983

  20. Roose, D., Caluwaerts, R.: Direct methods for the computation of a nonsimple turning point corresponding to a cusp. In: Numerical methods for bifurcation problems. T. Küpper, H.D. Mittelmann, H. Weber (eds.), pp. 426–440. ISNM, Vol. 70. Basel: Birkhäuser (1984)

    Google Scholar 

  21. Roose, D.: An algorithm for the computation of Hopf bifurcation points. In: Comparison with other methods. (To be published in J. Comp. Appl. Math.)

  22. Seydel, R.: Numerical computation of branch points in ordinary differential equations. Numer. Math.32, 51–68 (1979)

    Google Scholar 

  23. Seydel, R.: Numerical computation of branch points in nonlinear equations. Numer. Math.33, 339–352 (1979)

    Google Scholar 

  24. Spence, A., Jepson, A.: The numerical computation of turning points of nonlinear equations. In: Treatment of Integral Equations by numerical methods. C.T.H. Baker, G.F. Miller (eds.), pp. 169–183 London: Academic Press 1982

    Google Scholar 

  25. Spence, A., Werner, B.: Nonsimple turning points and cusps. IMA J. Numer. Anal.2, 413–427 (1982)

    Google Scholar 

  26. Spence, A., Jepson, A.: The numerical calculation of cusps, bifurcation points and isola formation points in two parameter problems. In: Numerical Methods for bifurcation problems. T. Küpper, H.D. Mittelmann, H. Weber (eds.), pp. 502–514. ISNM Vol. 70. Basel: Birkhäuser (1984)

    Google Scholar 

  27. Zeeman, E.C.: Bifurcation, Catastrophe and Turbulence. In: New Directions in Applied Mathematics. P.J. Hilton, G.S. Young (eds.), pp. 109–153. Berlin-Heidelberg-New York: Springer-Verlag 1982

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200 A, B-3030, Leuven, Belgium

    Dirk Roose & Robert Piessens

Authors
  1. Dirk Roose
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Piessens
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Roose, D., Piessens, R. Numerical computation of nonsimple turning points and cusps. Numer. Math. 46, 189–211 (1985). https://doi.org/10.1007/BF01390419

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01390419

Subject Classifications

Search

Navigation