Communications in Mathematical Physics

, Volume 305, Issue 1, pp 59–83 | Cite as

Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles

  • S. LiebscherEmail author
  • J. Härterich
  • K. Webster
  • M. Georgi


We consider cosmological models of Bianchi type. In particular, we are interested in the α-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity.

We prove the existence of a codimension-one family of solutions that limit, for t → −∞, onto a heteroclinic 3-cycle to the Kasner circle of equilibria. The theory extends to arbitrary heteroclinic chains that are uniformly bounded away from the three critical Taub points on the Kasner circle, in particular to all closed heteroclinic cycles of the Kasner map.


Homoclinic Orbit Stable Manifold Periodic Cycle Graph Transformation Heteroclinic Orbit 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BdSR86.
    Bugalho M.H., Rica da Silva A., Sousa Ramos J.: The order of chaos on a Bianchi-IX cosmological model. Gen. Rel. Grav. 18, 1263–1274 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. Beg10.
    Béguin F.: Aperiodic oscillatory asymptotic behavior for some bianchi spacetimes.[gr-qc], 2010
  3. HU09.
    Heinzle J.M., Uggla C.: Mixmaster: Fact and belief. Class. Quant. Grav. 26(7), 075016 (2009)MathSciNetADSCrossRefGoogle Scholar
  4. Mis69.
    Misner C.W.: Mixmaster universe. Phys. Rev. Lett. 22, 1071–1074 (1969)ADSzbMATHCrossRefGoogle Scholar
  5. Rin01.
    Ringström H.: The Bianchi IX attractor. Ann. Henri Poincaré 2(3), 405–500 (2001)ADSzbMATHCrossRefGoogle Scholar
  6. Rin09.
    Ringström H.: The Cauchy problem in general relativity. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society, 2009Google Scholar
  7. RT10.
    Reiterer M., Trubowitz E.: The BKL Conjectures for Spatially Homogeneous Spacetimes. [gr-qc], 2010
  8. SSTC98.
    Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O.: Methods of Qualitative Theory in Nonlinear Dynamics I. Volume 4 of Series on Nonlinear Science, Series A. Singapore World Scientific, (1998)Google Scholar
  9. WE05.
    Wainwright J., Ellis G.F.R. (eds.): Dynamical systems in cosmology. Cambridge: Cambridge University Press, 2005Google Scholar
  10. WH89.
    Wainwright J., Hsu L.: A dynamical systems approach to Bianchi cosmologies: orthogonal models of a class A. Class. Quant. Grav. 6(10), 1409–1431 (1989)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • S. Liebscher
    • 1
    Email author
  • J. Härterich
    • 2
  • K. Webster
    • 3
  • M. Georgi
    • 1
  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany
  2. 2.Fakultät für Mathematik Ruhr-UniversitätUniversitätsstr. 150BochumGermany
  3. 3.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations