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Abelian integrals and global hopf bifurcations

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Dynamical Systems and Bifurcations

Part of the book series: Lecture Notes in Mathematics ((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. Il'yashenko, Yu.S., Zeros of special Abelian integrals in a real domain, Funct. Anal. and Appl. 11 (1977), 309–311.

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  2. Keener, J.P., Infinite period bifurcation: and global bifurcation branches, SIAM J. Appl. Math. 41 (1981), 127–144.

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  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.

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  4. Griffiths, P. and Harris, J., Principles of Algebraic Geometry, J. Wiley & Sons, New York, 1978.

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  5. Rauch, H. and Lebowitz, A., Elliptic functions, Theta functions and Riemann surfaces, Williams and Wilkins, Baltimore, Maryland, 1973.

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  6. Cushman, R. and Sanders, J., A codimension two bifurcation with third order Picard-Fuchs equation (to appear in J. Diff. Eqns.)

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Vrije Universiteit, Amsterdam, the Netherlands

    J. A. Sanders

  2. Mathematics Institute, Rijksuniversiteit Utrecht, Utrecht, the Netherlands

    R. Cushman

Authors
  1. J. A. Sanders
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  2. R. Cushman
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Editor information

Boele L. J. Braaksma Hendrik W. Broer Floris Takens

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© 1985 Springer-Verlag

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Sanders, J.A., Cushman, R. (1985). Abelian integrals and global hopf bifurcations. In: Braaksma, B.L.J., Broer, H.W., Takens, F. (eds) Dynamical Systems and Bifurcations. Lecture Notes in Mathematics, vol 1125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075636

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  • DOI: https://doi.org/10.1007/BFb0075636

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15233-0

  • Online ISBN: 978-3-540-39411-2

  • eBook Packages: Springer Book Archive

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