Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-11T06:26:09.355Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic bifurcation and symmetry with an application to the Von Kármán equations

Published online by Cambridge University Press:  14 November 2011

André L. Vanderbauwhede
André L. Vanderbauwhede
Affiliation:
Instituut voor Theoretische Mechanika, Rijksuniversiteit Gent, Belgium Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, R.I., U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

A generic bifurcation theory is developed which is somewhat different from the approach in [4]. We put special emphasis on equations satisfying additional symmetry properties and on the non-generic bifurcation sets arising in this context. We apply our results on the von Kármán equations for the buckling of a rectangular plate under a compressive thrust and a normal load.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 3-4 , 1978 , pp. 211 - 235
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bancroft, S., Hale, J. K. and Sweet, D.Alternative problems for nonlinear functional equations. J. Differential Equations 4 (1968), 40–56.CrossRefGoogle Scholar
2Berger, M.On von Kármán's equations and the buckling of a thin elastic plate. I. The clamped plate. Comm. Pure Appl. Math. 20 (1967), 687–719.CrossRefGoogle Scholar
3Berger, M. and Fife, P. C.Von Kármán's equations and the buckling of a thin elastic plate. II. Plate with general edge conditions. Comm. Pure Appl. Math. 21 (1968), 227–241.CrossRefGoogle Scholar
4Chow, S., Hale, J. K. and Mallet-Paret, J.Applications of generic bifurcation. I. Arch. Rational Mech. Anal. 59 (1975), 159–188.CrossRefGoogle Scholar
5Chow, S., Hale, J. K. and Mallet-Paret, J.Applications of generic bifurcation. II. Arch. Rational Mech. Anal. 62 (1976), 209–235.CrossRefGoogle Scholar
6Golubitsky, M. and Guillemin, V.Stable Mappings and Their Singularities (New York: Springer, 1973).CrossRefGoogle Scholar
7Hale, J. K. Applications of alternative problems. CDS Lecture Notes 71–1, Brown University, 1971.Google Scholar
8Hale, J. K. and Rodrigues, H. M.Bifurcation in the Duffing equation with independent parameters. I. Proc. Roy. Soc. Edinburgh Sect. A. 87 (1977), 57–65.Google Scholar
9Hale, J. K. and Rodrigues, H. M.Bifurcation in the Duffing equation with independent parameters. II. Proc. Roy. Soc. Edinburgh Sect. A. 79 (1978), 317–326.CrossRefGoogle Scholar
10Knightly, G. and Sather, D.On nonuniqueness of solutions of the von Kármán equations. Arch. Rational Mech. Anal. 36 (1970), 65–78.CrossRefGoogle Scholar
11Knightly, G. H. Some mathematical problems from plate and shell theory. Proc. Michigan State Univ. Conf. Nonlinear Functional Anal. Differential Equations (New York: Marcel Dekker, 1976).Google Scholar
12Magnus, R. and Poston, T. On the full unfolding of the von Kármán equations at a double eigenvalue. Math. Rep. 109 (Geneva: Battelle, 1977).Google Scholar
13Mallet-Paret, J. Buckling of cylindrical shells with small curvature, preprint.Google Scholar
14Miller, W.Symmetry groups and their representations (New York: Academic Press, 1972).Google Scholar
15Poénaru, V. Singularites Cœ en présence de symétrie. Lecture Notes in Mathematics 510 (Berlin: Springer, 1976).Google Scholar
16Rodrigues, H. M. and Vanderbauwhede, A. L.Symmetric perturbations of nonlinear equations: symmetry of small solutions. Nonlin. Anal-Theory, Methods and Appl. 2 (1978), 27–46.CrossRefGoogle Scholar
17Ruelle, D.Bifurcations in the presence of symmetry group. Arch. Rational Mech. Anal. 51 (1973), 136–152.CrossRefGoogle Scholar
18Sattinger, D. H.Group representation theory and branch points of Nonlinear Functional Equations. SIAM J. Math. Anal. 8 (1977), 179–201.CrossRefGoogle Scholar
19Takens, F.Singularities of functions and vector fields. Nieuw Arch. Wisk. 20 (1972), 107–130.Google Scholar
20Thorn, R.Topological methods in biology. Topology 8 (1968), 313–335.Google Scholar
21Vainberg, M. M. and Trenogin, V. A.Theory of Branching of Solutions of Nonlinear Equations (Amsterdam: Noordhoff, 1974).Google Scholar
22Vanderbauwhede, A Generic and non-generic bifurcation for the von Kármán equations. J. Math. Anal. Appl., to appear.Google Scholar
23Vanderbauwhede, A.Alternative problems and symmetry. J. Math. Anal. Appl., 62 (1978), 483–494.CrossRefGoogle Scholar