Archive for Rational Mechanics and Analysis

, Volume 97, Issue 4, pp 321–352 | Cite as

On a four parameter family of planar vector fields

  • G. Dangelmayr
  • J. Guckenheimer
Article
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Keywords

Neural Network Complex System Vector Field Nonlinear Dynamics Electromagnetism 
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.
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References

  1. 1.
    A. Andronov, E. Leontovich, I. Gordon, & A. Maier, The Theory of Bifurcation of Plane Dynamical Systems, Wiley (New York) (1973).Google Scholar
  2. 2.
    I. Arnold, On the algebraic unsolvability of Lyapounov stability and the problem of classification of singular points of an analytic system of differential equations, Funct. Anal. Appl. 4, 173–180 (1970).Google Scholar
  3. 3.
    J. Boissonade & P. DeKepper, Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system, J. Phys. Chem. 84, 501–506 (1980).Google Scholar
  4. 4.
    F. Dumortier, R. Roussaire, & J. Sotomayor, Generic 3-parameter families of vector filelds on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3. Preprint.Google Scholar
  5. 5.
    J. Guckenheimer, Multiple Bifurcation Problems for Chemical Reactors, Physica D, 20, 1–20 (1986).Google Scholar
  6. 6.
    J. Guckenheimer, Global Bifurcations in Simple Models of a Chemical Reactor, SIAM Lectures in Appl. Math., 24, Part 2, 163–174 (1986).Google Scholar
  7. 7.
    J. Guckenheimer & P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag (New York), 453 pp. (1983).Google Scholar
  8. 8.
    Yu. S. Il'yashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funct. Anal. and Appl. 18, 199–209 (1984).Google Scholar
  9. 9.
    M. Medved, The unfoldings of a germ of vector fields in the plane with a singularity of codimension 3, Czech. Math. J. 35, 1–42 (1985).Google Scholar
  10. 10.
    I. Rehberg & G. Ahlers, Experimental observation of a codimension two bifurcation in a binary fluid mixture, Phys. Rev. Lett. 55, 500–503 (1985).Google Scholar
  11. 11.
    S. Schechter, The Saddle-Node Separatrix-Loop Bifurcation, preprint (1985).Google Scholar
  12. 12.
    F. Takens, Forced Oscillations and Bifurcations, Applications of Global Analysis, Communications of Maths. Institute, Rijksuniversiteit, Utrecht, 3, 1–59 (1974).Google Scholar
  13. 13.
    F. Takens, Unfoldings of certain singularities of vector fields: generalized Hopf bifurcations, J. Diff. Eq. 14, 476–493 (1973).Google Scholar
  14. 14.
    D. A. Vaganov, N. G. Samoilenko, & V. G. Abramov, Periodic regimes of continuous stirred tank reactors, Chem. Eng. Sci. 33, 1130–1140 (1978).Google Scholar
  15. 15.
    A. N. Varchenko, Estimate of the number of zeros of an abelian integral depending on a parameter and limit cycle, Funct. Anal. and Appl. 18, 99–107 (1984).Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1987

Authors and Affiliations

  • G. Dangelmayr
    • 1
    • 2
  • J. Guckenheimer
    • 1
    • 2
  1. 1.Universität TübingenGermany
  2. 2.Cornell UniversityUSA

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