Homoclinic Chaos in Relativistic Cosmology

  • Esteban Calzetta
Part of the NATO ASI Series book series (NSSB, volume 332)


We give a short review of Homoclinic Chaos, drawing instances of it from General Relativity, and putting emphasis on the application of Melnikov’s method for the detection of this kind of behavior. We describe in some detail two concrete manifestations of Homoclinic Chaos in relativistic Cosmology, one where the Universe may be described as a Hamiltonian dynamical system, and other where this is not possible, due to the presence of viscous matter. The overall implications of Chaos for Cosmology and General Relativity are discussed.


Lyapunov Exponent Cosmological Model Chaotic Behavior Integrable Hamiltonian System Quantum Cosmology 
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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  1. [1]
    Guckenheimer J. and Holmes P., 1983, Non-Linear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Berlin: Springer-Verlag).Google Scholar
  2. [2]
    Wiggins S., 1988, Global Bifurcations and Chaos (Heidelberg: Springer-Verlag).zbMATHCrossRefGoogle Scholar
  3. [3]
    Holmes, P., 1990, Poincaré, Celestial Mechanics, Dynamical Systems Theory and “Chaos” Phys. Rep. 193, 137.MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    Matinyan, S. G., Savvidi, G. K. and Ter-Arutyunyan-Savvidi, N. G., 1981, Classical Yang-Mills Mechanics, Nonlinear Color Oscillations Zh. Eksp. Teor. Fiz. 80, 830 (Engl. trans. Sov. Phys. JETP 53, 421).MathSciNetGoogle Scholar
  5. Kawabe, T. and Ohta, S., 1991, Order-to-Chaos Transition in SU(2) Yang-Mills Theory Phys. Rev. D 44, 1274.MathSciNetADSCrossRefGoogle Scholar
  6. Bambah, B. A., Lakshmibala, Mukku, C. and Sriram, M. S., 1993 Chaotic Behavior in Chern-Simons-Higgs Systems Phys. Rev. D 47, 4677.MathSciNetADSCrossRefGoogle Scholar
  7. [5]
    Chirikov B. V., 1979, A universal instability of many dimensional oscillator systems Phys. Rep. 52, 263.MathSciNetADSCrossRefGoogle Scholar
  8. [6]
    Reichl, L. E., and Zheng, W. M., 1987, Non Linear Resonance and Chaos in Conservative Systems, in Directions in Chaos ed. Hao Bai-Lin (Singapore, World Scientific).Google Scholar
  9. [7]
    Zaslavsky, G. M., Sagdeev, R. Z., Usikov, D. A. and Chernikov, A. A., 1991, Weak Chaos and Quasi Regular Patterns (Cambridge: Cambridge University Press).zbMATHCrossRefGoogle Scholar
  10. [8]
    Ornstein, D., 1974, Ergodic Theory, Randomness, and Dynamical Systems (New Haven, Yale University Press).zbMATHGoogle Scholar
  11. [9]
    Shields, P., 1974, The Theory of Bernoulli Shifts (Chicago: University of Chicago Press).Google Scholar
  12. [10]
    Arnold, V. I., 1978, Mathematical Methods of Classical Mechanics (Berlin: Springer-Verlag) (Second edition, 1989).zbMATHGoogle Scholar
  13. Arnold, V. I., Kozlov, V. V. and Neishtadt, A. I., 1988, Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems III, Encyclopaedia of Mathematical Sciences (Heidelberg: Springer-Verlag)CrossRefGoogle Scholar
  14. [11]
    Holmes, P. J. and Marsden, J. E., 1982, Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems J. Math. Phys. 23 669–75.MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. [12]
    Holmes, P. J. and Marsden, J. E., 1982, Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom Commun. Math. Phys. 82 523–44.MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. [13]
    Misner, C., 1972, Minisuperspace, in Magic without Magic, ed. J. Klauder (Freeman, San Francisco), p 441.Google Scholar
  17. [14]
    Palais, R. S., 1979 The Principle of Symmetric Criticality Commun. Math. Phys. 69, 19.MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. [15]
    Hu, B. L., 1982 Phys. Lett. 90A, 375.ADSGoogle Scholar
  19. [16]
    Arnold, V. I. and Avez, A., 1968, Ergodic Problems of Classical Mechanics (New York: Benjamin)Google Scholar
  20. [17]
    Regge, T. and Wheeler, J. A., 1957, Stability of a Schwarzschild singularity Phys. Rev. 108 1063–9MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. Vishveshwara, C. V., 1970, Stability of the Schwarzschild metric Phys. Rev. D 1 2870–9ADSCrossRefGoogle Scholar
  22. Moncrief, V., 1974, Gravitational perturbations of spherically symmetric systems. I. The exterior problem Ann. Phys. (N. Y.) 88 323–42MathSciNetADSCrossRefGoogle Scholar
  23. Chandrasekhar, S., 1983, The Mathematical Theory of Black Holes (Oxford: Clarendon Press).zbMATHGoogle Scholar
  24. [18]
    Bombelli, L. and Calzetta, E., 1992, Chaos around a Black Hole Class. Quantum Grav. 9, 2573.MathSciNetADSCrossRefGoogle Scholar
  25. [19]
    Calzetta, E. and El Hasi, C., 1993 Chaotic Friedmann-Robertson-Walker Cosmology Class. Quantum Grav. 10, 1825.ADSCrossRefGoogle Scholar
  26. [20]
    Calzetta E., El Hasi, C. and Tavakol, R, 1993, to appear Google Scholar
  27. [21]
    Ozorio de Almeida, A. M., 1988, Hamiltonian Systems, Chaos and Quantization (Cambridge: Cambridge University Press).zbMATHGoogle Scholar
  28. [22]
    Prigogine, I. and Eiskens, Y., 1987, Irreversibility, Stochasticity and Non Locality in Classical Dynamics, in Quantum Implications ed. B. J. Hiley and F. D. Peat (London: Routledge)Google Scholar
  29. [23]
    Sinai, Ya. G., 1970 Theory of Dynamical Systems, Lecture Notes Series 23 (Warsaw University)Google Scholar
  30. Cornfeld, I. P., Sinai, Ya. G. and Fomin, S. V., 1982, Ergodic Theory (Heidelberg: Springer-Verlag).zbMATHCrossRefGoogle Scholar
  31. [24]
    Khinchin, A., 1957, Mathematical Foundations of Information Theory (New York: Dover)zbMATHGoogle Scholar
  32. [25]
    Misner, C., Thorne, K. and Wheeler, A., 1972, Gravitation (San Francisco, Freeman).Google Scholar
  33. [26]
    Baierlein, R. F., Sharp, D. H. and Wheeler, J. A., 1962, Three Dimensional geometry as Carrier of Information about Time, Phys. Rev. 126, 1864.MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. [27]
    Wheeler, J. A., 1968, Super Space and the Nature of Quantum Geometrodynamics, in Batelle Rencontres ed. C. DeWitt and J. A. Wheeler (New York, Benjamin).Google Scholar
  35. [28]
    Ryan, M., 1972, Hamiltonian Cosmology (Berlin, Springer-Verlag)Google Scholar
  36. Ryan, M. and Shepley, L., 1975, Relativistic Homogeneous Cosmology (Princeton, Princeton University Press).Google Scholar
  37. [29]
    MacCallum, M. A. H., 1979, Anisotropic and Inhomogeneous Relativistic Cosmologies in Reneral Relativity ed. S. W. Hawking and W. Israel (Cambridge, Cambridge University Press) p. 533.Google Scholar
  38. [30]
    G. F. Smoot, 1993, COBE DMR Observations of the Early Universe, Class. Quantum Grav. 10 (1993).Google Scholar
  39. [31]
    Börner, G., 1988, The Early Universe (New York, Springer-Verlag) (2nd. Edition 1992).Google Scholar
  40. [32]
    Hawking, S. W., 1987, Quantum Cosmology, in 300 Years of Gravitation, ed. S. W. Hawking and W. Israel (Cambridge, Cambridge University Press) p. 631.Google Scholar
  41. [33]
    Parker, L. and Ford, L. H., 1977 Phys. Rev. D 16, 245MathSciNetADSCrossRefGoogle Scholar
  42. Parker, L. and Ford, L. H., 1977 Phys. Rev. D 16, 1601.MathSciNetADSCrossRefGoogle Scholar
  43. [34]
    Belinsky, V. A., Grishchuk, L. P., Khalatnikov, I. M. and Zel’dovich, Ya. B., 1985, Inflationary Stages in Cosmological Models with a Scalar Field, Phys. Lett. 155B, 232–6MathSciNetADSGoogle Scholar
  44. Belinsky, V. A., Grishchuk, L. P., Khalatnikov, I. M. and Zel’dovich, Ya. B., 1985 (same title) Zh. Eksp. Teor. Fiz. 89 346–60 (Engl. Trans. Sov. Phys. JETP 62 195-203)ADSGoogle Scholar
  45. Belinsky, V. A., Grishchuk, L. P., Khalatnikov, I. M. and Zel’dovich, Ya. B., 1985 (same title) Proceedings of the Third Seminar on Quantum Gravity ed. M. A. Markov, V. A. Berezin and V. P. Frolov (Singapore, World Scientific) 566–90Google Scholar
  46. Gottlober, S., Muller, V. and Starobinsky, A., 1991, Analysis of Inflation Driven by a Scalar Field and a Curvature Squared Term Phys. Rev. D. 43 2510–20.ADSCrossRefGoogle Scholar
  47. [35]
    Futamase, T., Rothman, T. and Matzner, R., 1989, Behavior of Chaotic Inflation in Anisotropic Cosmologies with Nonminimal Coupling Phys. Rev. D. 39 405–11ADSCrossRefGoogle Scholar
  48. Maeda, K., Stein-Schabes, J. and Futamase, T. 1989, Inflation in a Renormalizable Cosmological Model and the Cosmic No Hair Conjecture Phys. Rev. D 39 2848–53ADSCrossRefGoogle Scholar
  49. Amendola, L., Litterio, M. and Occhionero, F., 1990, The Phase Space View of Inflation (I) Int. J. Mod. Phys. A 5 3861–86MathSciNetADSCrossRefGoogle Scholar
  50. Demianski, M., 1991, Scalar Field, Nonminimal Coupling, and Cosmology Phys. Rev. D 44 3136–46MathSciNetADSCrossRefGoogle Scholar
  51. Demianski, M., de Ritis, R., Rubano, C. and Scudellaro, P. 1992 Scalar Fields and Anisotropy in Cosmological Models Phys. Rev. D 46 1391–8.MathSciNetADSCrossRefGoogle Scholar
  52. [36]
    Linde, A. D., 1982 Phys. Lett. 108B, 389.MathSciNetADSGoogle Scholar
  53. [37]
    Courant, R. and Hilbert, D., 1953, Methods of Mathematical Physics (New York, Wiley) Vol I, p. 531.Google Scholar
  54. [38]
    Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., 1985 Numerical Recipes: The Art of Scientific Computing (Cambridge, Cambridge University Press)zbMATHGoogle Scholar
  55. [39]
    Poincaré, H., 1892, Les Méthodes Nouvelles de la Mécanique Céleste (Gauthier-Villars, Paris).Google Scholar
  56. [40]
    Lichtenberg, A. J. and Lieberman, M. A., 1992 Regular and Chaotic Dynamics (New York, Springer-Verlag).zbMATHCrossRefGoogle Scholar
  57. [41]
    Pesin, Ya. B., 1977 Characteristic Lyapunov Exponents and Smooth Ergodic Theory, Uspekhi Mat. Nauk. 32:4, 55 (Engl. Trans. Russian Math. Surveys 32:4, 55).MathSciNetGoogle Scholar
  58. [42]
    Wolf, A., Swift, J., Swinney, H. and Vastano, J., 1985, Determining Lyapunov Exponents from a Time Series Physica 16D, 285–317.MathSciNetADSGoogle Scholar
  59. [43]
    Weinberg, S., 1989, The Cosmological Constant Problem Rev. Mod. Phys. 61, 1.MathSciNetADSzbMATHCrossRefGoogle Scholar
  60. [44]
    Gibbons, G. and Hawking, S. W., 1977, Phys. Rev. D 15, 2738.MathSciNetADSCrossRefGoogle Scholar
  61. Starobinsky, A. A., 1983, Piz’ma Zh. Eksp. Teor. Fiz. 37, 55 (Engl. Trans. JETP Letters 37, 66).ADSGoogle Scholar
  62. [45]
    Lieberman, M. A. and Tennyson, J. L., 1983, Chaotic motion along resonance layers in near integrable Hamiltonian systems with 3 or more degrees of freedom, Long-Time Prediction in Dynamics ed. CW Horton, LE Reichl and VG Szebehely (New York: John Wiley), p. 179.Google Scholar
  63. [46]
    Calzetta, E., 1991, Particle Creation, Inflation, and Cosmic Isotropy Phys. Rev. D 44, 3043.ADSCrossRefGoogle Scholar
  64. [47]
    Lyons, G., private communication.Google Scholar
  65. [48]
    Laflamme, R. and Shellard, E. P. S., 1987, Quantum Cosmology and Recollapse, Phys. Rev. D 35, 2315MathSciNetADSCrossRefGoogle Scholar
  66. Hawking, S. W., Laflamme, R. and Lyons, G. W., 1993, Origin of Time Asymmetry Phys. Rev. D 47, 5342.MathSciNetADSCrossRefGoogle Scholar
  67. [49]
    Weinberg, S., 1971, Entropy generation and the Survival of Proto-Galaxies in an Expanding Universe, Ap. J. 168, 175.ADSCrossRefGoogle Scholar
  68. [50]
    Weinberg, S., 1972 Gravitation and Cosmology (New York, John Wiley).Google Scholar
  69. [51]
    Jou, D., Casas-Vázquez, J. and Lebon, G., 1993, Extended Irreversible Thermodynamics (Berlin, Springer-Verlag).zbMATHCrossRefGoogle Scholar
  70. [52]
    Israel, W., 1988, Covariant Fluid Mechanics and Thermodynamics: an Introduction, in Relativistic Fluid Dynamics, ed. A. M. Anile and Y. Choquet-Bruhat (Berlin, Springer-Verlag).Google Scholar
  71. [53]
    Zakari, M. and Jou, D., 1993, Equations of State and Transport Equations in Viscous Cosmological Models, Phys. Rev. D 48, 1597.ADSCrossRefGoogle Scholar
  72. [54]
    Romano, R. and Pavón, D., 1993, Causal Dissipative Bianchi Cosmology Phys. Rev. D 47, 1396.ADSCrossRefGoogle Scholar
  73. Pavón, D. and Zimdahl, W., 1993, Dark Matter and Dissipation, Phys. Lett. 179A, 261.ADSGoogle Scholar
  74. [55]
    Shandarin, S. F. and Zeldovich, Ya. B., 1989, The Large Scale Structure of the Universe: Turbulence, Intermittency, Structures in a Self Gravitating Medium, Rev. Mod. Phys. 61, 185.MathSciNetADSCrossRefGoogle Scholar
  75. [56]
    Szebehely, V. G., 1983, Gravitational examples of non deterministic dynamics, Long-Time Prediction in Dynamics ed. CW Horton, LE Reichl and VG Szebehely (New York: John Wiley), p. 227.Google Scholar
  76. [57]
    Calzetta, E., 1986, The Behavior of the Efective Gravitational Constants for Broken SU(5), Ann. Phys. (N.Y.) 166, 214.ADSCrossRefGoogle Scholar
  77. [58]
    Simon, J. Z., 1990, Higher Derivative Lagrangians, Nonlocality, Problems, and Solutions, Phys. Rev. D 41, 3720MathSciNetADSCrossRefGoogle Scholar
  78. Parker, L. and Simon, J. Z., 1993, Einstein Equation with Quantum Corrections Reduced to Second Order, Phys. Rev. D 47, 1339.MathSciNetADSCrossRefGoogle Scholar
  79. [59]
    Calzetta, E. and Hu, B. L., 1989, Dissipation of Quantum Fields from Particle Creation, Phys. Rev. D 40, 656.MathSciNetADSCrossRefGoogle Scholar
  80. [60]
    Parker, L., 1968, Particle Creation in Expanding Universes Phys. Rev. Lett. 21, 562ADSCrossRefGoogle Scholar
  81. Parker, L., 1969, Quantized Fields and Particle Creation in Expanding Universes. I Phys. Rev. 183, 1057ADSzbMATHCrossRefGoogle Scholar
  82. Parker, L., 1971, Quantized Fields and Particle Creation in Expanding Universes. II Phys. Rev. D 3, 346.ADSCrossRefGoogle Scholar
  83. [61]
    Calzetta, E. and Sakellariadou, M., 1993, Semiclassical Effects and the Onset of Inflation, Phys. Rev. D 47, 3184.ADSCrossRefGoogle Scholar
  84. [62]
    Hu, B. L., Paz, J. P. and Zhang, Y., 1993, Quantum Origin of Noise and Fluctuations in Cosmology, Origin of Structure in the Universe, ed. E. Gunzig and P. Nardone (Dordrecht, Kluwer) p 227.CrossRefGoogle Scholar
  85. [63]
    E. Calzetta and B. L. Hu, 1993, Decoherence of Correlation Histories, in Directions in General Relativity, Vol. 2, ed. B. L. Hu and T. A. Jacobson (Cambridge, Cambridge University Press) p. 38.Google Scholar
  86. E. Calzetta and B. L. Hu, 1993, Noise and Fluctuations in Semiclassical Gravity, U. of Maryland preprint. Google Scholar
  87. [64]
    Hartle, J. B., 1993, The Quantum Mechanics of Closed Systems, in General Relativity and Gravitation 1992, ed. R. J. Gleiser, C. N. Kozameh and O. M. Moreschi (Bristol, IOP)p. 81.Google Scholar
  88. [65]
    Starobinsky, A. A., 1986, Stochastic De Sitter (Inflationary) Stage in the Early Universe, in Field Theory, Quantum gravity and Strings, ed. N. Sánchez and H. de Vega (Heidelberg, Springer-Verlag).Google Scholar
  89. [66]
    Brandenberger, R., Feldman, H., Mukhanov, V. and Prokopec, T., 1993, Gauge Invariant Cosmological Perturbations: Theory and Applications, in Origin of Structure in the Universe, ed. E. Gunzig and P. Nardone (Dordrecht, Kluwer) p 13.CrossRefGoogle Scholar
  90. [67]
    Cross, M. C. and Hohenberg, P. C., 1993, Pattern Formation outside of Equilibrium, Rev. Mod. Phys. 65, 851.ADSCrossRefGoogle Scholar
  91. [68]
    Vilenkin, A., 1983, Phys. Rev. D 27, 2848.MathSciNetADSCrossRefGoogle Scholar
  92. Halliwell, J. J., 1993, The Interpretation of Quantum Cosmological Models, in General Relativity and Gravitation 1992, ed. R. J. Gleiser, C. N. Kozameh and O. M. Moreschi (Bristol, IOP)p. 63.Google Scholar
  93. [69]
    Halliwell, J. J., 1987, Phys. Rev. D 36, 3626.MathSciNetADSCrossRefGoogle Scholar
  94. [70]
    Habib, S., 1990, Classical Limit in Quantum Cosmology: Quantum Mechanics and the Wigner Function, Phys. Rev. D 42, 2566.MathSciNetADSCrossRefGoogle Scholar
  95. [71]
    Paz, J. P. and Sinha, S., 1991, Decoherence and Back Reaction: the Origin of the Semiclassical Einstein Equations, Phys. Rev. D 44, 1038.ADSCrossRefGoogle Scholar
  96. Hu, B. L., Paz, J. P. and Sinha, S., 1993, Minisuperspace as a Quantum Open System, in Directions in General Relativity, Vol. 1, ed. B. L. Hu, M. P. Ryan and C. V. Viveshwara (Cambridge, Cambridge University Press) p. 145.CrossRefGoogle Scholar
  97. [72]
    Berry, M., 1983, Semiclassical mechanics of regular and irregular motion, Chaotic Behavior of Deterministic Systems ed. G Iooss, RHG Helleman and R Stora (New York: North-Holland), p. 171.Google Scholar
  98. [73]
    Ashtekar, A., 1991, Lectures on Non Perturbative Canonical Gravity, (Singapore, World Scientific).zbMATHGoogle Scholar
  99. [74]
    Ashtekar, A. and Pullin, J., 1990, Bianchi Cosmologies, a new Description, Proc. Israel Phys. Soc. 9, 65.MathSciNetGoogle Scholar
  100. [75]
    Capovilla, R., Dell, J. and Jacobson, T., 1993, The Initial Value Problem in Light of Ashtekar’s Variables, in Directions in General Relativity, Vol. 2, ed. B. L. Hu and T. A. Jacobson (Cambridge, Cambridge University Press) p. 66.Google Scholar
  101. [76]
    Temam, R., 1988, Infinite Dimensional Dynamical Systems in Mechanics and Physics (New York, Springer-Verlag).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Esteban Calzetta
    • 1
    • 2
  1. 1.IAFEBuenos AiresArgentina
  2. 2.Physics DepartmentUniversity of Buenos AiresArgentina

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