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Abelian integrals and global hopf bifurcations

  • J. A. Sanders
  • R. Cushman
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1125)

Abstract

We give a detailed and simplified proof of a theorem of Yu. S. Il'yashenko which concerns the uniqueness of certain limit cycles. A slightly extended version of this theorem is then applied to a global Hopf bifurcation problem treated by Keener.

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References

  1. 1.
    Il'yashenko, Yu.S., Zeros of special Abelian integrals in a real domain, Funct. Anal. and Appl. 11 (1977), 309–311.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Keener, J.P., Infinite period bifurcation: and global bifurcation branches, SIAM J. Appl. Math. 41 (1981), 127–144.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Il'yashenko, Yu.S., The multiplicity of limit cycles arising from perturbations of the form w′ = P2/Q1 of a Hamiltonian equation in the real and complex domain, Trudy Sem. Petrovsk 3 (1978), 49–60 = AMS Transl. 118 (1982), 191–202.MathSciNetGoogle Scholar
  4. 4.
    Griffiths, P. and Harris, J., Principles of Algebraic Geometry, J. Wiley & Sons, New York, 1978.zbMATHGoogle Scholar
  5. 5.
    Rauch, H. and Lebowitz, A., Elliptic functions, Theta functions and Riemann surfaces, Williams and Wilkins, Baltimore, Maryland, 1973.zbMATHGoogle Scholar
  6. 6.
    Cushman, R. and Sanders, J., A codimension two bifurcation with third order Picard-Fuchs equation (to appear in J. Diff. Eqns.)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • J. A. Sanders
    • 1
  • R. Cushman
    • 2
  1. 1.Department of Mathematics and Computer ScienceVrije UniversiteitAmsterdamthe Netherlands
  2. 2.Mathematics InstituteRijksuniversiteit UtrechtUtrechtthe Netherlands

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