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Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case

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Dynamical Systems and Turbulence, Warwick 1980

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 898))

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References

  1. Andronov, A.A., Leontovich, E.A., Gordon, I.I., & Maier, A.G.: Theory of bifurcations of dynamical systems on a plane. Israel Progr. for Sc. Transl. Ltd. 1971.

    Google Scholar 

  2. Arnold, V.I.: Lectures on bifurcations and versal familics. In: Russ. Math. Surv. 27, 54–123 (1972).

    Google Scholar 

  3. Birkhoff, G.D.: Dynamical systems. In: AMS. Coll. Publ. 9 New York: Amer. Math. Soc. 1927. Reprinted 1966.

    Google Scholar 

  4. Borel, A.: Linear algebraic groups. (Notes by H. Bass). Benjamin 1979.

    Google Scholar 

  5. Broer, H.W.: Bifurcations of singularities in volume preserving vector fields. Ph.D. thesis, Groningen 1979.

    Google Scholar 

  6. Broer, H.W.: Quasi periodic flow near a codimension one singularity of a divergence free vector field in dimension three. This volume.

    Google Scholar 

  7. Brunovksy, P.: One parameter families of diffeomorphisms. In: Symposium on Differential Equations and Dynamical Systems, pp 29–33. Springer 1971.

    Google Scholar 

  8. Dumortier, F.: Singularities of vector fields. In: Monografias de Mathemàtica 32, IMPA.

    Google Scholar 

  9. Dumortier, F., Rodrigues, P.R., & Roussarie, R.: Germs of diffeomorphisms in the plane. To be published.

    Google Scholar 

  10. Ebin, D., & Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. In: Ann. of Math. 92, 102–163 (1970).

    Google Scholar 

  11. Ŝil'nikov, L.P.: A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle focus type. In: Math. USSR Sbornik, 10, 91–102 (1970).

    Google Scholar 

  12. Gérard, R., & Levelt, A.H.M.: Sur les connexions a singularités régulières dans le cas de plusiers variables. In: Funkcialaj Elevacioj 19, 149–173 (1976).

    Google Scholar 

  13. Hartman, P.: Ordinary differential equations. Wiley & Sons 1964.

    Google Scholar 

  14. Landau L.D., & Lifshitz, E.M.: Fluid mechanics, Pergamon 1959.

    Google Scholar 

  15. Mather, J.N.: Stratifications and mappings. In: Dynamical Systems, ed. M.M. Peixoto, pp. 195–232. Acad. Press 1973.

    Google Scholar 

  16. J. Moser.: Stable and random motions in dynamical systems. Princeton Univ. Press, 1973.

    Google Scholar 

  17. Narasimhan, R.: Analysis on real and complex manifolds. North Holland 1968.

    Google Scholar 

  18. Palis, J. & Takens, F.: Topological equivalence in normally hyperbolic dynamical systems. In: Topology 16, 335–345 (1977).

    Google Scholar 

  19. Poènaru, V.: Singularités C∞ en présence de symmétrie. Springer 1976.

    Google Scholar 

  20. Poincaré, H.: Thèse. In: Oeuvres 1, pp. LIX-CXXIX (1879). Gauthiers Villars 1928.

    Google Scholar 

  21. Robinson, R.C.: Generic properties of conservative systems I, II. In: Amer. J. Math., 92, 562–603, 879–906 (1970).

    Google Scholar 

  22. Ruelle, D., & Takens, F.: On the nature of turbulence. In: Comm. Math. Phys. 20, 167–192 (1971).

    Google Scholar 

  23. Siegel, C.L.: Über die analytische Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung. In: Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. 21–30 (1952).

    Google Scholar 

  24. Sotomayor, J.: Generic one-parameter families of vector fields on two-dimensional manifolds. In: Publ. Math. IHES 43, 5–46 (1973).

    Google Scholar 

  25. Sternberg, S.: On the structure of local homeomorphisms of Euclidean n-space III. In: Amer. J. Math. 81, 578–604 (1959).

    Google Scholar 

  26. Takens, F.: Homoclinic points in conservative systems. In: Inventiones Math. 18, 267–292 (1972).

    Google Scholar 

  27. Takens, F.: A nonstabilisable jet of a singularity of a vector field. In: Dynamical Systems, ed. M.M. Peixoto, pp. 583–597, Acad. Press 1973.

    Google Scholar 

  28. Takens, F.: Forced oscillations and bifurcations. In: Applications of Global Analysis I, Comm. of the Math. Inst. Rijksuniversiteit Utrecht (1974).

    Google Scholar 

  29. Takens, F.: Singularities of vector fields. In: Publ. Math. IHES 43, 48–100 (1974).

    Google Scholar 

  30. Varadarajan, V.S.: Lie groups, Lie algebras and their representations. Prentice Hall 1974.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Groningen University, P.O. Box 800, Groningen, The Netherlands

    Henk Broer

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  1. Henk Broer
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David Rand Lai-Sang Young

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© 1981 Springer-Verlag

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Broer, H. (1981). Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case. In: Rand, D., Young, LS. (eds) Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics, vol 898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091907

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  • DOI: https://doi.org/10.1007/BFb0091907

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11171-9

  • Online ISBN: 978-3-540-38945-3

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