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Bifurcation from a manifold

  • David Chillingworth
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1463)

Keywords

Implicit Function Theorem Height Function Cambridge Philos Cusp Point Simple Singularity 
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References

  1. [A]
    Arnol’d, V.I., Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups and singular projections of smooth surfaces, Russian Math. Surveys 34 (1979), 1–42.CrossRefzbMATHGoogle Scholar
  2. [ACZE]
    Ambrosetti, A., Coti Zelati, V. and Ekeland, I., Symmetry breaking in Hamiltonian systems, J. Diff. Eqns. 67 (1987), 165–184.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BL]
    Bröcker, Th. and Lander, L., Differentiable germs and catastrophes, London Math. Soc. Lecture Notes 17, Cambridge University Press 1975.Google Scholar
  4. [B1]
    Bruce, J.W., Canonical stratifications: the simple singularities, Math. Proc. Cambridge Philos. Soc. 88 (1980), 265–272.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [B2]
    Bruce, J.W., The duals of generic hypersurfaces, Math. Scand. 49 (1981), 36–60.MathSciNetzbMATHGoogle Scholar
  6. [B3]
    Bruce, J.W., On singularities, envelopes and elementary differential geometry, Math. Proc. Cambridge Philos. Soc. 89 (1981), 43–48.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [B4]
    Bruce, J.W., Generic reflections and projections of surfaces, Math. Scand. 54 (1984), 262–278.MathSciNetzbMATHGoogle Scholar
  8. [B5]
    Bruce, J.W., Seeing — the mathematical viewpoint, Math. Intelligencer 6 (1984), 18–25.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [B6]
    Bruce, J.W., Geometry of singular sets, Math. Proc. Cambridge Philos. Soc. 106 (1989), 495–509.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [BG1]
    Bruce, J.W. and Giblin, P.J., On real simple singularities, Math. Proc. Cambridge Philos. Soc. 88 (1980), 273–279.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [BG2]
    Bruce, J.W. and Giblin, P.J., Curves and singularities, Cambridge University Press 1984.Google Scholar
  12. [C1]
    Chillingworth, D.R.J., Bifurcation from an orbit of symmetry, in Singularities and Dynamical Systems, S.N. Pnevmatikos (ed.), pp. 285–294, North-Holland, Amsterdam 1985.Google Scholar
  13. [C2]
    Chillingworth, D.R.J., The ubiquitous astroid, in Proc. Int. Symp. on The Physics of Structure Formation: Theory and Simulation, Institut für Informationsverarbeitung, Universität Tübingen, 1986, W. Güttinger and G. Dangelmayr (eds.), pp. 372–386, Springer-Verlag, Berlin 1987.CrossRefGoogle Scholar
  14. [CMW]
    Chillingworth, D.R.J., Marsden, J.E. and Wan, Y.-H., Symmetry and bifurcation in three-dimensional elasticity, Part I: Arch. Rat. Mech. Anal. 80 (1982), 295–331, Part II: Arch. Rat. Mech. Anal. 8 (1983), 363–395.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [CH]
    Chow, S.-N. and Hale, J.K., Methods of bifurcation theory, Springer-Verlag, New York, 1982.CrossRefzbMATHGoogle Scholar
  16. [D1]
    Dancer, E.N., On the existence of bifurcating solutions in the presence of symmetries, Proc. Roy. Soc. Edinburgh 85A (1980), 321–336.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [D2]
    Dancer, E.N., The G-invariant implicit function theorem in infinite dimensions, Proc. Roy. Soc. Edinburgh 92A (1982), 13–30.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [D3]
    Dancer, E.N., Perturbation of zeros in the presence of symmetries, J. Austral. Math. Soc. (Ser. A) 36 (1984), 106–125.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [GG]
    Golubitsky, M. and Guillemin, V., Stable mappings and their singularities, Springer-Verlag, New York 1973.CrossRefzbMATHGoogle Scholar
  20. [GS]
    Golubsky, M. and Schaeffer, D., Imperfect bifurcation in the presence of symmetry, Common. Math. Phys. 67 (1979), 205–232.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [GSS]
    Golubitsky, M., Stewart, I. and Schaeffer, D.G., Singularities and Groups in Bifurcation Theory, Vols. I, II., Springer-Verlag, New York, 1985, 1988.CrossRefzbMATHGoogle Scholar
  22. [HT]
    Hale, J.K. and Taboas, P.Z., Interaction of damping and forcing in a second order equation, Nonlin. Anal. Th. Meth. Appl. 2 (1978), 77–84.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [L]
    Looijenga, E.J.N., Structural stability of smooth families of C functions, Thesis, Univ. Amsterdam, 1974.Google Scholar
  24. [M]
    Marsden, J.E., Qualitative methods in bifurcation theory, Bull. Amer. Math. Soc. 84 (1978), 1125–1148.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Ma]
    Mather, J.N., Generic projections, Ann. Math. (2), (1973), 226–245.Google Scholar
  26. [Mg]
    Magnus, R.J., On perturbations of a translationally invariant differential equation, Proc. Roy. Soc. Edinburgh 110A (1988), 1–25.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [R]
    Reeken, M., Stability of critical points under small perturbations, Part I: Topological theory, Manuscripta Math. 7 (1972), 387–411.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [V1]
    Vanderbauwhede, A., Symmetry and bifurcations near families of solutions, J. Diff. Eqns. 36 (1980), 173–187.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [V2]
    Vanderbauwhede, A., Local Bifurcation and Symmetry, Research Notes in Math. 75, Pitman, London 1982.zbMATHGoogle Scholar
  30. [Wa]
    Wall, C.T.C., Geometric properties of generic differentiable manifolds, in: Geometry and Topology III, Lecture Notes in Math. 597, Springer-Verlag, Berlin 1976.Google Scholar
  31. [We]
    Weinstein, A., Perturbation of periodic manifolds of Hamiltonian systems, Bull. Amer. Math. Soc. 77 (1971), 814–818.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [WM]
    Wan, Y.-H. and Marsden, J.E., Symmetry and bifurcation in three-dimensional elasticity, Part III, Arch. Rat. Mech. Anal. 84 (1983), 203–233.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • David Chillingworth

There are no affiliations available

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