Archive for Rational Mechanics and Analysis

, Volume 99, Issue 4, pp 301–328 | Cite as

Stability and folds

  • J. H. Maddocks


It is known that when one branch of a simple fold in a bifurcation diagram represents (linearly) stable solutions, the other branch represents unstable solutions. The theory developed here can predict instability of some branches close to folds, without knowledge of stability of the adjacent branch, provided that the underlying problem has a variational structure. First, one particular bifurcation diagram is identified as playing a special role, the relevant diagram being specified by the choice of functional plotted as ordinate. The results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram. In many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph. The theory is particularly useful in applications where a solution branch has a succession of folds.

The theory is illustrated with applications to simple models of thermal selfignition and of a chemical reactor, both of which systems are of Émden-Fowler type. An analysis concerning an elastic rod is also presented.


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  1. Antman, S. S., & G. Rosenfeld (1978), Global behavior of buckled states of nonlinearly elastic rods. SIAM Review 20, p. 513. Corrigenda (1980) 22, p. 186.Google Scholar
  2. Chandrasekhar, S. (1939), An Introduction to the Study of Stellar Structure. U. of Chicago Press.Google Scholar
  3. Chow, S. N., & R. Lauterbach (1985), A Bifurcation Theorem for Critical Points of Variational Problems. IMA Report # 179.Google Scholar
  4. Crandall, M. C., & P. H. Rabinowitz (1973), Bifurcation, Perturbation of Simple Eigenvalues and Linearized Stability. Archive for Rational Mechanics and Analysis 52, pp. 160–192.Google Scholar
  5. Davis, H. T. (1962), Introduction to Nonlinear Differential and Integral Equations, Dover, New York.Google Scholar
  6. Decker, D. W., & H. B. Keller (1980), Multiple Limit Point Bifurcation. J. of Math. Anal. and Appl. 65, pp. 417–430.Google Scholar
  7. Decker, D. W., & H. B. Keller (1981), Path Following Near Bifurcation. Comm. Pure and Applied Math. 34, p. 149.Google Scholar
  8. émden, R. (1907), Gaskugeln. Berlin and Leipzig.Google Scholar
  9. Ericksen, J. L. (1975), Equilibrium of bars. J. of Elasticity 5, pp. 191–201.Google Scholar
  10. Gel'fand, I. M. (1963), Some problems in the theory of quasilinear equations. Amer. Math. Soc. Transl. Series 2 Vol. 29, pp. 295–381.Google Scholar
  11. Iooss, G., & D. D. Joseph (1980), Elementary Stability and Bifurcation Theory. Springer, New York.Google Scholar
  12. Jackson, R. (1973), A simple geometric condition for instability in catalyst pellets at unit Lewis number. Chemical Engineering Science 28, pp. 1355–1358.Google Scholar
  13. James, R. D. (1981), The equilibrium and post-bucking behavior of an elastic curve governed by a non-convex energy. J. Elasticity 11, pp. 239–269.Google Scholar
  14. Joseph, D. D. (1979), Factorization Theorems and Repeated Branching of Solutions at a Simple Eigenvalue. Annals of the New York Academy of Sciences 316, pp. 150–167.Google Scholar
  15. Joseph, D. D., & T. S. Lundgren (1973), Quasilinear Dirichlet Problems Driven by Positive Sources. Arch. Rational Mech. Anal. 49, pp. 241–269.Google Scholar
  16. Katz, J. (1978) (1979), On the number of unstable modes of an equilibrium I and II. Mon. Not. R. Astr. Soc. 183, pp. 765–769, and 189, pp. 817–822.Google Scholar
  17. Keller, H. B. (1977), Constructive Methods for Bifurcation and Nonlinear Eigenvalue Problems. Proc. 3rd Int. Symp. on Computing Method in Applied Science and Engineering.Google Scholar
  18. Maddocks, J. H. (1984), Stability of nonlinearly elastic rods. Archive for Rational Mechanics and Analysis 85, pp. 311–354.Google Scholar
  19. Maddocks, J. H. (1985), Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles. SIAM J. of Math. Anal. 16, pp. 47–68.Google Scholar
  20. Maddocks, J. H., & Jepson, A. D. (1986), Exchange of stability in variational bifurcation problems. In preparation.Google Scholar
  21. Mehta, B. N., & R. Aris (1971), A note on the form of the Émden-Fowler equation. J. Math. Anal. & Applications 36, pp. 611–621.Google Scholar
  22. Sattinger, D. H. (1972), Stability of Solutions of Nonlinear Equations. J. Math. Anal. & Appl. 39, pp. 1–12.Google Scholar
  23. Shatah, J., & W. Strauss (1985), Instability of nonlinear bound states. Comm. Math. Phys. 100, pp. 173–190.Google Scholar
  24. Thompson, J. M. T. (1979), Stability predictions through a succesion of folds. Phil. Trans. Roy. Soc. of London 192 A, p. 1386.Google Scholar
  25. Weinberger, H. (1978), On the Stability of Bifurcating Solutions. Reprint from “Nonlinear Analysis” (dedicated to Erich Rothe), Academic Press.Google Scholar
  26. Wente, Henry C. (1980), The Stability of the Axially Symmetric Pendent Drop. Pac J. of Math. 88, pp. 421–469.Google Scholar
  27. Zeidler, Eberhard (1984), Nonlinear Functional Analysis and its Applications, Part III. Variational Methods and Optimization, Springer-Verlag, New York.Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • J. H. Maddocks
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege Park

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