Hopf Bifurcation for an Infinite Delay Functional Equation

  • Olof J. Staffans
Conference paper
Part of the NATO ASI Series book series (volume 37)

Abstract

In this lecture we discusse a Hopf bifurcation problem for the functional equation
$$ x(t) = F(\alpha ,x_t ),t \in R. $$
(1.1)
Here x is a vector in R n, the parameter α is an element of a finite dimensional real Banach space A, and F is a mapping from Α × BUC(R;R n) into R n. Moreover, F(α, 0) = 0, so x ≡ 0 is a solution of (1.1). In addition we suppose that the linearization of (1.1) has a one-parameter family of nontrivial periodic solutions at a critical value α 0 of the parameter. Our aim is to show that also the nonlinear equation has a oneparameter family of nontrivial periodic solutions for some values of α close to α 0.

Keywords

Manifold Convolution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 35 (1971), 312–348.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    S-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin and New York, 1982.Google Scholar
  3. 3.
    J. R. Claeyssen, Effect of delays on functional differential equations, J. Differential Equations 20 (1976), 404–440.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    J. R. Claeyssen, The integral-averaging bifurcation method and the general one-delay equation, J. Math. Anal. Appl. 78 (1980), 429–439.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    O. Diekmann and S. A. van Gils, Invariant manifolds for Volterra integral equations of convolution type, J. Differential Equations 54 (1984), 139–180.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    R. R. Goldberg, Fourier Transforms, Cambridge University Press, London, 1970.Google Scholar
  7. 7.
    G. Gripenberg, Periodic solutions of an epidemic model, J. Math. Biology 10 (1980), 271–280.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    G. Gripenberg, On some epidemic models, Quart. Appl. Math. 39 (1981), 317327.Google Scholar
  9. 9.
    G. Gripenberg, Stability of periodic solutions of some integral equations, J. reine angew. Math. 331 (1982), 16–31.MathSciNetMATHGoogle Scholar
  10. 10.
    J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin and New York, 1975.Google Scholar
  11. 11.
    J. K. Hale, Nonlinear oscillations in equations with delays, in Nonlinear Oscillations in Biology, Lectures in Applied Mathematics, Vol. 17., 157–189, American Mathematical Society, Providence, 1978.Google Scholar
  12. 12.
    J. K. Hale and J. C. F. de Oliveira, Hopf bifurcation for functional equations, J. Math. Anal. Appl. 74 (1980), 41–58.Google Scholar
  13. 13.
    G. S. Jordan, O. J. Staffans and R. L. Wheeler, Local analyticity in weighted L1-spaces and applications to stability problems for Volterra equations, Trans. Amer. Math. Soc. 274 (1982), 749–782.MathSciNetMATHGoogle Scholar
  14. 14.
    G. S. Jordan, O. J. Staffans and R. L. Wheeler, Convolution operators in a fading memory space: The critical case, SIAM J. Math. Analysis, to appear.Google Scholar
  15. 15.
    G. S. Jordan, O. J. Staffans and R. L. Wheeler, Subspaces of stable and unstable solutions of a functional differential equation in a fanding memory space: The critical case, to appear.Google Scholar
  16. 16.
    J. C. F. de Oliveira, Hopf bifurcation for functional differential equations, Nonlinear Anal. 4 (1980), 217–229.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    J. C. F. de Oliveira and J. K. Hale, Dynamic behavior from bifurcation equations, Tôhoku Math. J. 32 (1980), 577–592.MATHGoogle Scholar
  18. 18.
    O. J. Staffans, On a neutral functional differential equation in a fading memory space, J. Differential Equations 50 (1983), 183–217.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    O. J. Staffans, Hopf bifurcation of functional and functional differential equations with infinite delay, to appear.Google Scholar
  20. 20.
    H. W. Stech, Hopf bifurcation calculations for functional differential equations, J. Math. Anal. Appl. 109 (1985), 472–491.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    H. W. Stech, Nongeneric Hopf bifurcations in functional differential equations, SIAM J. Math. Anal. 16 (1985), 1134–1151.MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Olof J. Staffans
    • 1
  1. 1.Institute of MathematicsHelsinki University of TechnologyEspoo 15Finland

Personalised recommendations