Advertisement

A Dynamical Systems Approach to the Oscillatory Singularity in Bianchi Cosmologies

  • P. K-H. Ma
  • J. Wainwright
Part of the NATO ASI Series book series (NSSB, volume 332)

Abstract

We describe the behaviour of the orthogonal Bianchi cosmologies of types VIII and IX near the big-bang in terms of an attractor of a dynamical system. Comparisons are made with previous work.

Keywords

Equilibrium Point Bianchi Type Type VIII Einstein Field Equation Dynamical System Approach 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anosov, D.V. and Arnold, V.I. (1988), Dynamical Systems I, Springer-Verlag, New York, NY.zbMATHCrossRefGoogle Scholar
  2. Barrow, J.D. (1979), Nature, 272, 211.ADSCrossRefGoogle Scholar
  3. Barrow, J.D. (1982), Phys. Rep., 85, 1.MathSciNetADSCrossRefGoogle Scholar
  4. Belinskii, V.A. and Khalatnikov, I.M. (1969), Sov. Phys. JETP, 29, 911.ADSGoogle Scholar
  5. Belinskii, V.A., Khalatnikov, I.M. and Lifshitz, E.M., (1970), Adv. Phys., 19, 525.ADSCrossRefGoogle Scholar
  6. Bogoyavlensky, O.I., and Novikov, S.P. (1973), Sov. Phys. JETP, 37, 747.ADSGoogle Scholar
  7. Bogoyavlensky, O.I. (1985), Methods in the Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics, Springer Verlag, New York, NY.zbMATHCrossRefGoogle Scholar
  8. Burd, A.B., Buric, N. and Ellis, G.F.R. (1990), Gen. Rel. Grav., 22, 349.MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. Ellis, G.F.R. and MacCallum, M.A.H. (1969), Commun. Math. Phys., 12, 108–141.MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. Francisco, G. and Matsas, G.E.A. (1988), Gen. Rel. Grav., 20, 1047.MathSciNetADSCrossRefGoogle Scholar
  11. Grebogi, C., Ott, E., Pelikan, S., and Yorke, J.A. (1984), Physica D, 13, 251.MathSciNetADSCrossRefGoogle Scholar
  12. Hobill, D., Bernstein, D., Simkins, D. and Welge, M. (1989), in Proc. 12th Int. Conf. Gen. Rel. Grav., Boulder, Univ. of Colorado, abstracts p. 337.Google Scholar
  13. Khalatnikov, I.M. and Pokrovsky, V.L. (1972), in Magic without Magic, (ed. J. Klauder), W. H. Freeman, San Francisco, CA.Google Scholar
  14. Khalatnikov, I.M., Lifshitz, E.M., Khanin, K.M., Shchur, L.N., and Sinai, Ya. G. (1985), J. Stat. Phys., 38, 97.MathSciNetADSCrossRefGoogle Scholar
  15. Kramer, D., Stephani, H., MacCallum, M.A.H. and Herlt, E. (1980), Exact Solutions of Einstein’s Field Equations, Deutscher Verlag der Wissenschaften, Berlin, and Cambridge University Press, Cambridge, UK.zbMATHGoogle Scholar
  16. Ma, P.K-H. (1988), A Dynamical Systems Approach to the Oscillatory Singularity in Cosmology, M. Math. thesis, University of Waterloo.Google Scholar
  17. MacCallum, M.A.H. (1979), in Physics of the Expanding Universe, (ed. M. Demianski), Lecture Notes in Physics, Vol. 109, Springer-Verlag, Berlin and Heidelburg.Google Scholar
  18. Milnor, J. (1985), Commun. Math. Phys., 99, 177.MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. Misner, C.W. (1969), Phys. Rev. Lett, 22, 1071.ADSzbMATHCrossRefGoogle Scholar
  20. Misner, C.W. (1970), in Relativity, (eds., M. Carmelli, S. Fickler, and L. Witten), Plenum Publishing, New York, NY.Google Scholar
  21. Nemytskii, V.V. and Stepanov, V.V. (1960), Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NY.zbMATHGoogle Scholar
  22. Ryan, M. P., Jr. and Shepley, L.C. (1975), Homogeneous Relativistic Cosmologies, Princeton Univ. Press, Princeton, NJ.Google Scholar
  23. Shimada, I. and Nagashima, T. (1979), Prog. Theoret. Phys., 61, 1605.MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. Taub, A. (1951), Ann. Math., 53, 472.MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. Wainwright, J. and Hsu, L. (1989), Class. Quantum Grav., 6, 1409.MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. Zardecki, A. (1983), Phys. Rev., D28, 1235.MathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • P. K-H. Ma
    • 1
  • J. Wainwright
    • 1
  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations