Linear response of the Lorenz system

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 66, Issue 3, 2002, Pages

  Reick, Christian H ^a  

^a ALFRED WEGENER INSTITUTE FOR POLAR AND MARINE RESEARCH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

CHEMICAL BONDS; DIFFUSION; ELASTIC MODULI; HYDROCARBONS; LIGHT SCATTERING; MICROEMULSIONS; VECTORS;

COLLECTIVE DIFFUSION; DYNAMIC LIGHT SCATTERING (DLS); SELF-DIFFUSION; VISCOELASTIC SYSTEMS;

VISCOELASTICITY;

ARTICLE;

EID: 33646984047     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.66.036103     Document Type: Article

References (37)

1. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
2. See, e.g., D.N. Subarev, Statistische Thermodynamik des Nichtgleichgewichts (Akademie-Verlag, Berlin, 1976), and references therein.
3. D. Ruelle, Commun. Math. Phys. 187, 227 (1997).
4. D. Ruelle, Phys. Lett. A 245, 220 (1998).
5. G. Gallavotti, J. Stat. Phys. 84, 899 (1996);
6. G. GallavottiJ. Math. Phys. 41, 4061 (2000).
7. S.V. Ershov, Phys. Lett. A 177, 180 (1993).
8. J.D. Farmer, Phys. Rev. Lett. 55, 351 (1985).
9. K. Takahashi, A. Ichimura, H. Hirooka, and N. Saito, J. Phys. Soc. Jpn. 54, 500 (1985).
10. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Applied Mathematical Sciences Vol. 42 (Springer, New York, 1983).
11. R. Abraham and J.E. Marsden, Foundations of Mechanics (Benjamin, Reading, MA, 1978).
12. D. Ruelle, J. Stat. Phys. 95, 393 (1999).
13. E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
14. C. Sparrow, The Lorenz Equations (Springer, New York, 1982).
15. S. Grossmann, Z. Phys. B: Condens. Matter 57, 77 (1984).
16. S. Grossmann and S. Thomae, Z. Naturforsch. 32a, 1553 (1977).
17. W. Breymann and H. Thomas, in From Phase Transitions to Chaos, edited by G. Györgyi, I. Kondor, L. Sasvary, and T. Tel (World Scientific, Singapore, 1992).
18. M. Falcioni, S. Isola, and A. Vulpiani, Phys. Lett. 144A, 341 (1990);
19. G.F. Carnevale, M. Falcioni, S. Isola, R. Purini, and A. Vulpiani, Phys. Fluids A 3, 2247 (1991).
20. L. Biferale, I. Daumont, G. Lacorata, and A. Vulpiani, Phys. Rev. E 65, 016302 (2001).
21. N.G. van Kampen, Phys. Norv. 5, 10 (1971).
22. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II (Springer, Berlin, 1985), pp. 196–199.
23. J.F.C. van Velsen, Phys. Rep. 41, 135 (1978);
24. M. Bianucci, R. Mannella, X. Fan, P. Grigolini, and B.J. West, Phys. Rev. E 47, 1510 (1993);
25. Phys. Rev. EM. Bianucci, R. Mannella, B.J. West, and P. Grigolini, 50, 2630 (1994);
26. M. Falcioni and A. Vulpiani, Physica A 215, 481 (1995);
27. M. Bianucci, R. Mannella, and P. Grigolini, Phys. Rev. Lett. 77, 1258 (1996).
28. H. Suhl, Physica B 199/200, 1 (1994).
29. N. Saito and Y. Matsunaga, J. Phys. Soc. Jpn. 58, 3089 (1989).
30. C.H. Reick, Math. Comput. Simul. 40, 281 (1996).
31. Similar results as those presented here have also been obtained for (Formula presented), where the Lorenz system is also chaotic.
32. All integrations of the Lorenz system were done with a fourth order Runge-Kutta integration scheme with time step (Formula presented).
33. For frequencies around (Formula presented) very long integration times T (up to (Formula presented) are needed to see the beginning of the plateau. The reason for this is that in this range of “internal frequencies” the chaotic background (Formula presented) is very large so that the first term in Eq. (2.10) gives considerable contributions for smaller T.
34. Why for small values of (Formula presented) the phase (Formula presented) seems to lock to (Formula presented) or (Formula presented) has not further been analyzed; for the present purpose only the stabilization at sufficiently large (Formula presented) is of interest.
35. B. Eckhardt and G. Ott, Z. Phys. B: Condens. Matter 93, 259 (1994).
36. H. Haken, Phys. Lett. 53A, 77 (1975).
37. M.Y. Li, Tin Win, C.O. Weiss, and N.R. Heckenberg, Opt. Commun. 80, 119 (1990), and references therein.