Relativistic chaos is coordinate invariant

Physical Review Letters

Volumn 91, Issue 23, 2003, Pages

  Motter, Adilson E ^a  

^a MAX PLANCK INSTITUT FÜR PHYSIK KOMPLEXER SYSTEME   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ASTROPHYSICS; CHAOS THEORY; ITERATIVE METHODS; LYAPUNOV METHODS; MATHEMATICAL TRANSFORMATIONS; PHASE SPACE METHODS; PROBLEM SOLVING; VECTORS;

COORDINATE SYSTEMS; SPACE DIFFEOMORPHISMS;

RELATIVITY;

EID: 0346686286     PISSN: 00319007     EISSN: 10797114     Source Type: Journal    
DOI: 10.1103/PhysRevLett.91.231101     Document Type: Article

References (28)

1. See, for example, L. Bombelli and E. Calzetta, Classical Quantum Gravity 9, 2573 (1992); 10.1088/0264-9381/9/12/004
2. W.M. Vieira and P.S. Letelier, Phys. Rev. Lett. 76, 1409 (1996).10.1103/PhysRevLett.76.1409
3. J.D. Barrow, Phys. Rep. 85, 1 (1982).10.1016/0370-1573(82)90171-5
4. Deterministic Chaos in General Relativity, edited by D. Hobill, A. Burd, and A. Coley (Plenum, New York, 1994).
5. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, England, 1994).
6. V.I. Oseledec, Trans. Moscow Math. Soc.TMMSD4 19, 197 (1968).
7. C.W. Misner, Phys. Rev. Lett. 22, 1071 (1969); 10.1103/PhysRevLett.22.1071
8. V.A. Belinskii, I.M. Khalatnikov, and E.M. Lifshitz, Adv. Phys. 19, 525 (1970).
9. G. Francisco and G.E.A. Matsas, Gen. Relativ. Gravit.GRGVA8 20, 1047 (1988);
10. A.B. Burd, N. Buric, and G.F.R. Ellis, Gen. Relativ. Gravit.GRGVA8 22, 349 (1990);
11. B.K. Berger, Gen. Relativ. Gravit.GRGVA8 23, 1385 (1991);
12. D. Hobill et al, Classical Quantum Gravity 8, 1155 (1991).10.1088/0264-9381/8/6/013
13. M. Szydlowski, Phys. Lett. A 176, 22 (1993).10.1016/0375-9601(93)90311-M
14. S.E. Rugh, in Ref.�
15. N.J. Cornish and J.J. Levin, Phys. Rev. D 55, 7489 (1997).10.1103/PhysRevD.55.7489
16. G. Contopoulos, B. Grammaticos, and A. Ramani, J. Phys. AJPHAC5 27, 5357 (1994).10.1088/0305-4470/27/15/031
17. J.D. Barrow and J. Levin, Phys. Rev. Lett. 80, 656 (1998).10.1103/PhysRevLett.80.656
18. A.E. Motter and P.S. Letelier, Phys. Lett. A 285, 127 (2001).10.1016/S0375-9601(01)00349-8
19. R.D. Ramey and N.L. Balazs, Found. Phys.FNDPA4 31, 371 (2001).10.1023/A:1017598705383
20. B.R. Fayad, Ergod. Theor. Dyn. Syst. 22, 437 (2002).
21. N.J. Cornish, gr-qc/9602054;
22. N.J. Cornishgr-qc/9709036.
23. We exclude the cases where the measure accumulates on trivial subsets of the invariant set, such as isolated points and simple boundaries
24. The orbits in the extended phase space will be unbounded along (Formula presented). When the transformed system depends only periodically on (Formula presented), this can be remedied by taking (Formula presented) rather than (Formula presented) as the new phase-space coordinate, where (Formula presented) is the period of the time dependence. Lyapunov exponents can also be defined for some nonautonomous systems [J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985)].10.1103/RevModPhys.57.617
25. A.E. Motter and P.S. Letelier, Phys. Rev. D 65, 068502 (2002).10.1103/PhysRevD.65.068502
26. C. Grebogi, E. Ott, and J. Yorke, Phys. Rev. A 37, 1711 (1988).10.1103/PhysRevA.37.1711
27. A.E. Motter (to be published).
28. When the relevant invariant sets are saddles rather than attractors, the numerical computation of Lyapunov exponents on these sets might be a highly nontrivial task. This technical problem can be overcome, however, with the manifestly invariant fractal techniques of Ref.�