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Spontaneous Symmetry Breaking in Bifurcation Problems

  • D. H. Sattinger

Abstract

In analyzing the dynamics of a physical system governed by nonlinear equations the following questions occur: Are there equilibrium states of the system? How many are there? Are they stable or unstable? What happens as external parameters are varied? In particular, what happens when a known solution becomes unstable as some parameter passes through a critical value?

Keywords

Symmetry Breaking Hopf Bifurcation Bifurcation Point Spherical Shell Spontaneous Symmetry Breaking 
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.
    Auchmuty, J. F. G. and Nicolis, G., “Dissipative Structures, Catastrophes, and Pattern Formation: A Bifurcation Analysis,” Proc. Nat. Acad. Sci. USA, 71 (1974), 2748–2751.MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Birman, J. L., “Symmetry change in continuous phase transitions in crystals,” Second International Colloquium on Group Theory in Physics, (Nijmegan, 1972) ed. by A. Janner.Google Scholar
  3. 3.
    Busse, F., “Patterns of Convection in Spherical Shells,” J. Fluid Mech. (1975), 72, 67–85.ADSMATHCrossRefGoogle Scholar
  4. 4.
    Cowan, J. D. and Ermentrout, G. B., “Secondary Bifurcation in Neural Nets,” SIAM Journal Applied Mathematics, to appear.Google Scholar
  5. 5.
    Cowan, J. D. and Ermentrout, G. B., “A mathematical theory of visual hallucination patterns,” Biological Cybernetics, to appear.Google Scholar
  6. 6.
    Dicke, R. H., Phys. Rev. 93 (1954), p. 99.ADSMATHCrossRefGoogle Scholar
  7. 7.
    Erneux, T. and Herschkowitz-Kaufman, T. and Herschkowitz-Kaufman, “Rotating waves as asymptotic solutions of a model chemical reaction,” Jour. Chem. Phys. 66 (1977), 248–250.ADSGoogle Scholar
  8. 8.
    Erneux, T. and Herschkowitz-Kaufman, “The Bifurcation Diagram of Model Chemical Reactions,” Annals of the New York Academy of Sciences, 316.Google Scholar
  9. 9.
    Fife, P., “Pattern formation in reacting and diffusing systems,” Jour. Chem. Phys. 64 (1976), 554–564.ADSGoogle Scholar
  10. 10.
    Green, Luks, and Kozak, “A precise determination of the critical exponent γ for the YBG square-well fluid,” to appear.Google Scholar
  11. 11.
    Haken, H. “Laser Theory,” in Handbuch der Physik, vol. XXV/2C Springer-Verlag, Berlin, 1970.Google Scholar
  12. 12.
    Hopf, E., “Abzweigung linear periodischer Lösung eines Differential Systems,” Berichte der Math.-Phys. Klasse der Sâchsischen Akademie der Wissenschaften zu Leipzig XCIV (1942) 1–22.Google Scholar
  13. 13.
    Hepp, K. and Lieb, E. H., “Phase transitions in reservoir-driven open systems with applications to lasers and superconductors,” Helvetica Physica Acta, 46 (1973), 574–603.Google Scholar
  14. 14.
    Jaric, M. V. and Birman, J. L., “New algorithms for the Molien functions,” Jour. Math. Phys. 18 (1977), 1456–1458.MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Kozak, Rice, and Weeks, Rice, and Weeks, “Analytic approach to the theory of phase transitions,” J. Chem. Phys. 52 (1970), 2416.Google Scholar
  16. 16.
    Koschmieder, E. L., “Benard Convection,” Advances in Chemical Physics 26 (1974), 177–212.CrossRefGoogle Scholar
  17. 17.
    Lax, M., “Phase transitions and superfluidity,” Brandeis Lectures, 1956, (Gordon and Breach, N.Y. (1968)).Google Scholar
  18. 18.
    Michel, L. and Radicati, L. A., “The geometry of the octet, ” Ann. Inst. Henri Poincaré Sect. A. Physique, Vol. 18 (1973), 185–214.MathSciNetMATHGoogle Scholar
  19. 19.
    Michel, L. and Radicati, L. A., “Properties of the breaking of hadronic internal symmetry,” Annals of Physics 66 (1971), 758–783.MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Raveche, H. J., and Stuart, C., “Towards a molecular theory of freezing,” J. C.em. Phys. 63 (1975), 136–152.Google Scholar
  21. 21.
    Roberts, P. H., “Dynamo Theory,” in Mathematical Problems in the Geophysical Sciences, Lectures in Applied Mathematics, X IV, Amer. Math. Soc., Providence, 1971.Google Scholar
  22. 22.
    Sattinger, D. H., “Group Theoretic Methods in Bifurcation Theory,” Lecture Notes in Mathematics, #762, Springer-Verlag, 1979.Google Scholar
  23. 23.
    Sattinger, D. H., “Group representation theory, bifurcation theory, and pattern formation,” Jour. Functional Analysis 28 (1978).Google Scholar
  24. 24.
    Sattinger, D. H., “Selection rules for pattern formation,” Arch. Rat. Mech. Anal. Rat. Mech. Anal. 66 (1977), 31–42.MathSciNetADSMATHGoogle Scholar
  25. 25.
    Sattinger, D. H., “Bifurcation from rotationally invariant states,” Jour. Math. Phys. 19 (1978), 1720–1732.MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Turing, A. M., “The chemical basis of morphogenesis,” Philo. Trans. Roy. Soc. London B237 (1952), 37–72.ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • D. H. Sattinger
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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