Numerische Mathematik

, Volume 97, Issue 3, pp 537–554 | Cite as

On necessary and sufficient conditions for numerical verification of double turning points

  • Ken’ichiro Tanaka
  • Sunao Murashige
  • Shin’ichi Oishi


This paper describes numerical verification of a double turning point of a nonlinear system using an extended system. To verify the existence of a double turning point, we need to prove that one of the solutions of the extended system corresponds to the double turning point. For that, we propose an extended system with an additional condition. As an example, for a finite dimensional problem, we verify the existence and local uniqueness of a double turning point numerically using the extended system and a verification method based on the Banach fixed point theorem.


Nonlinear System Point Theorem Additional Condition Turning Point Fixed Point Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kanzawa, Y., Oishi, S.: Calculating bifurcation points with guaranteed accuracy. IEICE Trans. Fundamentals E82-A~6, 1055–1061 (1999)Google Scholar
  2. 2.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Second Edition, Springer-Verlag, New York, 1998Google Scholar
  3. 3.
    Oishi, S.: Numerical verification of existence and inclusion of solutions for nonlinear operator equations. J. Comput. Appl. Math. 60, 171–185 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Oishi, S.: Introduction to Nonlinear Analysis. Corona-sha, Tokyo, 1997 [in Japanese]Google Scholar
  5. 5.
    Roose, D., Piessens, R.: Numerical computation of nonsimple turning points and cusps. Numer. Math. 46, 189–211 (1985)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Spence, A., Werner, B.: Nonsimple turning points and cusps. IMA J. Numer. Anal. 2, 413–427 (1982)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Tanaka, K.: Computer-assisted existence proof of the bifurcation points of Duffing’s equation. Batchelor thesis of the Department of Mathematical Engineering, Faculty of Engineering, the University of Tokyo, 2002 [in Japanese]Google Scholar
  8. 8.
    Yang, Z.-H., Keller, H.B.: A direct method for computing higher order folds. SIAM J. Sci. Stat. Comput. 7, 351–361 (1986)Google Scholar
  9. 9.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications I – Fixed-Point Theorems. Springer, New York, 1986Google Scholar
  10. 10.
    Zeidler, E.: Applied Functional Analysis – Main Principles and Their Applications. Applications to Mathematical Sciences 109, Springer, New York, 1995Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ken’ichiro Tanaka
    • 1
  • Sunao Murashige
    • 2
  • Shin’ichi Oishi
    • 3
  1. 1.Department of Mathematical InformaticsGraduate School of Information Science and TechnologyJapan
  2. 2.Department of Complexity Science and EngineeringGraduate School of Frontier SciencesJapan
  3. 3.Department of Computer ScienceSchool of Science and EngineeringTokyoJapan

Personalised recommendations