Extensions of an algorithm for the analysis of nongeneric Hopf bifurcations, with applications to delay-difference equations

  • Jeffery M. Franke
  • Harlan W. Stech
Research Articles
Part of the Lecture Notes in Mathematics book series (LNM, volume 1475)


A previously derived algorithm for the analysis of the Hopf bifurcation in functional differential equations is extended, allowing the elementary approximation of an existence and stability — determining scalar bifurcation function. With the assistance of the symbolic manipulation program MACSYMA [5], [9] this algorithm is used to implement the algorithm and to investigate the nature of nongeneric Hopf bifurcations in scalar delay — difference equations.


Hopf Bifurcation Functional Differential Equation Negative Real Part Integrodifferential Equation Bifurcation Structure 
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  1. 1.
    Aboud, N. (1988): Contributions to the Computer-Aided Analysis of Functional Differential Equations. Master's Thesis, University of MinnesotaGoogle Scholar
  2. 2.
    Aboud, N., Sathaye, A. Stech H. (1988): BIFDE: Software for the Investigation of the Hopf Bifurcation Problem in Functional Differential Equations. Proceedings of the 27th Conference on Decision and Control, IEEE, 821–824Google Scholar
  3. 3.
    Chow, S.-N, Mallet-Paret, J. (1977): Integral averaging and Hopf bifurcation. J. Differential Equations, 26, 112–159MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Claeyssen, J. R. (1980): The integral-averaging bifurcation method and the general one-delay equation. J. Math. Anal. Appl, 78, 429–439MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Drinkard, R. D., Sulinski N. (1981): MACSYMA: A Program For Computer Algebraic Manipulation (Demonstrations and Analysis). Naval Underwater Systems Center Technical Document 6401, Reprinted by SYMBOLICSGoogle Scholar
  6. 6.
    Franke, J. (1989): Symbolic Hopf Bifurcations for Functional Differential Equations. Master's Thesis, University of MinnesotaGoogle Scholar
  7. 7.
    Hale, J. K. (1971): Functional Differential Equations. Applied Math. Sci., Vol. 3, Springer-Verlag, New YorkzbMATHGoogle Scholar
  8. 8.
    Hassard, B., Kazarinoff, N., Wan, Y-H. (1981): Theory and Applications of Hopf Bifurcation. London Math. Soc. Lecture Notes, No. 41, Cambridge University Press, CambridgezbMATHGoogle Scholar
  9. 9.
    MACSYMA Reference Manual, Version 11 (1986), prepared by the MACSYMA group of SYMBOLICS. Inc. 11 Cambridge Center, Cambridge, MA 02142Google Scholar
  10. 10.
    Marsden, J. E., McCracken, M. (1976): The Hopf Bifurcation and its Applications. Applied Math. Sciences, Vol. 19, Springer-Verlag, New YorkzbMATHGoogle Scholar
  11. 11.
    Sathaye, A. (1986): BIFDE: A Numerical Software Package for the Hopf Bifurcation Problem in Functional Differential Equations. Master's Thesis, Virginia Polytechic Institute and State UniversityGoogle Scholar
  12. 12.
    Stech, H. W.: Generic Hopf bifurcations for a class of integro-differential equations. SubmittedGoogle Scholar
  13. 13.
    Stech, H. W. (1985): Hopf bifurcation calculations for functional differential equations. Journal of Math Analysis and Applications, 109, No. 2, 472–491MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Stech, H. W. (1985): Nongeneric Hopf Bifurcations in Functional Differential Equations. SIAM J. Appl. Math., 16, No. 6, 1134–1151MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Jeffery M. Franke
  • Harlan W. Stech

There are no affiliations available

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