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Asymptotic Poincaré compactification and finite-time singularities

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Abstract

We provide an extension of the method of asymptotic decompositions of vector fields with finite-time singularities by applying the central extension technique of Poincaré to the dominant part of the vector field at approach to the singularity. This leads to a bundle of fan-out asymptotic systems whose equilibria at infinity govern the dynamics of the asymptotic solutions of the original system. We show how this method can be useful to describe a single-fluid isotropic universe at the time of maximum expansion, and discuss possible relations of our results to structural stability and non-compact phase spaces.

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References

  1. Y. Choquet-Bruhat and S. Cotsakis, J. Geom. Phys. 43, 345 (2002); gr-qc/0201057.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. S. Cotsakis, Gen Rel. Grav. 36, 1183 (2004); grqc/ 0309048.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. S. Cotsakis and I. Klaoudatou, J. Geom. Phys. 55, 306 (2005); gr-qc/0409022.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. S. Cotsakis and I. Klaoudatou, J. Geom. Phys. 57, 1303 (2007); gr-qc/0604029.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. S. Cotsakis, Talking about singularities, in Proceedings of MG11 Meeting on General Relativity (World Scientific, 2009); gr-qc/0703084.

    Google Scholar 

  6. J. D. Barrow, Class. Quantum Grav. 21, L79 (2004), gr-qc/0403084; see also J. D. Barrow, Class. Quantum Grav. 21, 5619 (2004); J. D. Barrow, S. Cotsakis, and A. Tsokaros, Class. Quantum Grav. 27, 165017 (2010).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. S. Nojiri, S. D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005); hep-th/0501025.

    Article  ADS  Google Scholar 

  8. S. Cotsakis and G. Kittou, Phys. Lett. B 712, 16 (2012).

    Article  ADS  Google Scholar 

  9. S. Cotsakis and A. Tsokaros, Phys. Lett. B 651, 341 (2007); S. Cotsakis, G. Kolionis, and A. Tsokaros, Phys. Lett. B 721, 1-6 (2013).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. I. Antoniadis, S. Cotsakis, and I. Klaoudatou, Class. Quantum Grav. 27, 235018 (2010); arXiv: 1010.6175.

    Article  MathSciNet  ADS  Google Scholar 

  11. I. Antoniadis, S. Cotsakis, and I. Klaoudatou, Forschr. Phys. 61, 20 (2013); arXiv: 1206.0090.

    Article  MathSciNet  ADS  Google Scholar 

  12. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, and R. Sundrum, Phys. Lett. B 480, 193 (2000); hepth/ 0001197.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. R. Penrose, Conformal treatment of infinity, in Relativity, Groups and Topology, C. DeWitt and B. DeWitt eds. (Gordon and Breach, New York, 1964), pp. 563–584.

    Google Scholar 

  14. I. Bendixson, Acta Math. 24, 1 (1901).

    Article  MathSciNet  Google Scholar 

  15. S. Cotsakis and J. D. Barrow, The dominant balance at cosmological singularities, J. Phys. Conf. Ser. 68, 012004 (2007), gr-qc/0608137.

    Article  Google Scholar 

  16. S. Lefschetz, Differential Equations: Geometric Theory (2nd Ed., Dover, 1977).

    Google Scholar 

  17. L. Perko, Differential Equations and Dynamical Systems (3rd Ed., Springer, 2001).

    Book  MATH  Google Scholar 

  18. F. Dumortier, J. Llibre, and J. C. Artés, Qualitative Theory of Planar Differential Systems (Springer, 2006).

    MATH  Google Scholar 

  19. J. D. Meiss, Differential Dynamical Systems (SIAM, 2007).

    Book  MATH  Google Scholar 

  20. J. Wainwright and G. F. R. Ellis, Dynamical Systems in Cosmology (CUP, 1997).

    Book  Google Scholar 

  21. S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-Time (CUP, 1973).

    Book  MATH  Google Scholar 

  22. Y. Choquet-Bruhat, General Relativity and the Einstein Equations (OUP, 2009).

    MATH  Google Scholar 

  23. S. Cotsakis, Structure of infinity in cosmology (to appear in IJMPD), arXiv: 1212.6737.

  24. E. A. González Velasco, Trans. Amer. Math. Soc. 143 201 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Cima, F. Manosas, and J. Villadelprat, Topology 37, 261 (1998).

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Research Group of Geometry, Dynamical Systems and Cosmology University of the Aegean, Karlovassi, 83 200, Samos, Greece

    Spiros Cotsakis

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  1. Spiros Cotsakis
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Correspondence to Spiros Cotsakis.

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Cotsakis, S. Asymptotic Poincaré compactification and finite-time singularities. Gravit. Cosmol. 19, 240–245 (2013). https://doi.org/10.1134/S0202289313040099

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  • DOI: https://doi.org/10.1134/S0202289313040099

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