Abstract
The effect of external random forces on the static and dynamic behavior of the Lorenz model is investigated. Results of a numerical calculation in the conductive, convective, and turbulent regimes are reported. The properties of static and time-dependent correlation functions of the three degrees of freedom of the model are analyzed for varying strength of the external noise level and compared with the behavior of the unforced system.
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References
E. N. Lorenz,J. Atmos. Sci. 20:130 (1963).
R. N. May,Nature 261:459 (1976);Science 186:645 (1974); J. A. Meyer,J. Phys. (Paris)C5:29 (1978).
J. B. McLaughlin and P. C. Martin,Phys. Rev. A 12:186 (1973).
J. B. McLaughlin,J. Stat. Phys. 15:307 (1976); K. A. Robbins,Math. Proc. Cambridge Phil. Soc. 82:309 (1977); J. H. Curry,Commun. Math. Phys. 60:192 (1978); and Bounded Solutions of Finite Dimensional Approximations to the Boussinesq Equations,SIAM J. Math. Anal. (to appear); H. Yahata.Prog. Theor. Phys. Suppl. 64:176 (1978);Prog. Theor. Phys. 61:791 (1979); C. Boldrighini and V. Franceschini,Commun. Math. Phys. 64:159 (1979); O. E. Rössler,Phys. Lett. 57A:397 (1976);71A:155 (1979); P. Coullet, C. Tresser, and A. Arnéodo,Phys. Lett. 72A:268 (1979).
K.-I. Nakamura.Prog. Theor. Phys. 57:1874 (1977),59:64 (1978);Prog. Theore. Phys. Suppl 64:378 (1978);Proc. Inst. Nat. Sci. Nihon Univ. 14:9 (1979).
H. Haken,Synergetics, 2nd ed. (Springer, Heidelberg, 1978).
M. Hénon,Commun. Math. Phys. 50:69 (1976);J. B. McLaughlin, Phys. Lett. 72A:271 (1979); J. H. Curry,Commun. Math. Phys. 68:129 (1979); F. R. Marotto,Commun. Math. Phys. 68:187 (1979).
J. Guckenheimer, G. Oster, and A. Ipaktchi,J. Math. Biol. 4:101 (1977).
D. Ruelle and F. Takens,Commun. Math. Phys. 20:167 (1971); R. F. Williams, inTurbulence Seminar, Lecture Notes in Mathematics, Vol. 615 (Springer, New York, 1977), p. 94; J. Guckenheimer, inThe Hopf Bifurcation and its Application, Applied Mathematical Sciences, Vol. 19 (Springer, New York, 1976), p. 368; D. Ruelle, inMathematical Problems in Theoretical Physics, Lecture Notes in Physics, Vol. 80 (Springer, New York, 1978), p. 341.
M. Hénon and Y. Pomeau, inTurbulence and Navier Stokes Equation, Lecture Notes in Mathematics, Vol. 565 (Springer, New York, 1976), p. 29; O. Lanford, inTurbulence Seminar, Lecture Notes in Mathematics, Vol. 615 (Springer, New York, 1977), p. 113.
S. D. Feit,Commun. Math. Phys. 61:249 (1978).
T. Shimizu and N. Morioka,Phys. Lett. 66A:182, 447 (1978);69A:148 (1978).
V. Franceschini and A. Feigenbaum, Sequence of Bifurcations in the Lorenz Model, preprint (1979).
K. Matsuno,Phys. Rev. A 11:1016 (1975).
I. Shimada and T. Nagashima,Prog. Theor. Phys. 59:1033 (1978);61:1605 (1979); T. Nagashima,Prog. Theor. Phys. Suppl. 64:368 (1978).
Tien-Yien Li and J. A. Yorke,Am. Math. Mon. 82:985 (1975); Y. Oono,Prog. Theor. Phys. 59:1028 (1978).
G. Jooss,J. Phys. (Paris)C5:99 (1978).
J. L. Kaplan and J. A. Yorke,Commun. Math. Phys. 67:93 (1979); J. A. Yorke and E. D. Yorke,J. Stat. Phys. 21:263 (1979).
M. J. Feigenbaum,J. Stat. Phys. 19:25 (1978); and Los Alamos preprint No. LA-UR-78-1155; B. Derrida,J. Phys. (Paris)C5:49 (1978); A. Arnéodo, P. Coullet, and C. Tresser,Phys. Lett. 70A:74 (1979).
J. B. McLaughlin, Ph.D. thesis, Harvard University (1974).
M. Lücke,J. Stat. Phys. 15:455 (1976).
B. Sonneborn-Schmick, Diplomthesis, Universität Marburg (1977).
E. Knobloch,J. Stat. Phys. 20:695 (1979).
Y. Aizawa and J. Shimada,Prog. Theor. Phys. 57:2146 i1977).
F. C. Hoppensteadt and J. M. Hyman,SIAM J. Appl. Math. 32:73 (1977); S. Grossmann and S. Thomae,Z. Naturforsch. 32A:1353 (1977); H. Fujisaka and T. Yamada,Z. Naturforsch. 33A:1455 (1978); T. Kai and K. Tomita, preprint (1979).
Y. Ueda, C. Hayashi, and N. Akamatsu,Electron. Commun. Jpn. 56A:27 (1973); Y. Ueda,J. Stat. Phys. 20:181 (1979); J. E. Flaherty and F. C. Hoppensteadt,Stud. Appl. Math. 58:5 (1975); G. M. Zaslavskky,Phys. Lett. 69A:145 (1978); A. Ito,Prog. Theor. Phys. 61:815 (1979); K. Tomita and T. Kai,J. Stat. Phys. 21:65 (1979), and references cited therein.
K. Kitahara, W. Horsthemke, and R. Lefever,Phys. Lett. 70A:377 (1979), and references cited therein.
R. Kraichnan,J. Fluid Mech. 5:497 (1959); P. C. Martin, E. D. Siggia, and H. A. Rose,Phys. Rev. A 8:423 (1973).
B. Saltzman:J. Atmos. Sci. 19:329 (1962).
G. Bennettin, M. Casartelli, L. Galgani, A. Giorgelli, and J. M. Strelchyn,Nuovo Cimento 44B:183 (1978);50B:211 (1979).
K. Takeyama,Prog. Theor. Phys. 60:613 (1978).
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Zippelius, A., Lücke, M. The effect of external noise in the Lorenz model of the Bénard problem. J Stat Phys 24, 345–358 (1981). https://doi.org/10.1007/BF01013305
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DOI: https://doi.org/10.1007/BF01013305