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A Dynamical Systems Approach to Nonlinear Stellar Pulsations

Published online by Cambridge University Press:  12 April 2016

J. R. Buchler
J. R. Buchler*
Affiliation:
University of Florida, Gainsville, FL32611, USA

Abstract

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Over the last decade we have seen the application of novel techniques to the old problem of nonlinear stellar pulsations. Together with numerical hydrodynamics this approach provides a more fundamental understanding of the systematics of the pulsational behavior. For weakly nonadiabatic pulsations, whether regular or multi-periodic, dimensional reduction techniques lead to amplitude equations and to a description in terms of modal interactions and resonances. In particular they shed new light on the bump progression in the classical Cepheids. In more dissipative stars numerical hydrodynamical modelling has uncovered the existence of irregular variability, both in radiative and in convective models. An application of modern dynamical systems techniques has shown that this behavior occurs according to well understood routes from regular to chaotic behavior. The mechanism is very robust and represents the first non ad hoc theoretical explanation of irregular stellar variability. Finally, we discuss how a comparison with observations of irregular variability shows the need for more suitable observations, on the one hand, and of better techniques of signal processing, on the other.

Information

Type
I. Fundamental Theories
Information
International Astronomical Union Colloquium , Volume 134: Nonlinear Phenomena in Stellar Variability , 1993 , pp. 9 - 31
Copyright
Copyright © Kluwer 1993

References

Aikawa, T.: 1987, Astrophysics and Space Science 139, 281.Google Scholar
Aikawa, T.: 1988, Astrophysics and Space Science 149, 149.Google Scholar
Aikawa, T.: 1990, Astrophysics and Space Science 164, 295.Google Scholar
Arp, H.C. 1955, Astronomical Journal 60, 1.CrossRefGoogle Scholar
Baker, N.H.: 1966, in Stellar Evolution, eds. Stein, R.F. and Cameron, A.G.W. (New York: Plenum Press), p. 333.Google Scholar
Baker, N., Moore, D. and Spiegel, E.A.: 1966, Astronomical Journal 71, 845.Google Scholar
Barranco, M., Buchler, J.R., and Livio, M.: 1981, Astrophysical Journal 242, 1226.Google Scholar
Barranco, M., Buchler, J.R., and Regev, O.: 1982, Astrophysics and Space Science 84, 463.Google Scholar
Bergé, P., Pomeau, Y. and Vidal, C.: 1986, Order Within Chaos (N. Y.: Wiley).Google Scholar
Buchler, J.R.: 1985, in Chaos in Astrophysics, NATO ASI Ser. C, Vol. 161, eds. Buchler, J.R., Perdang, J.M. and Spiegel, E.A. (Dordrecht: Reidel), p. 137.Google Scholar
Buchler, J.R.: 1990, in Nonlinear Astrophysical Fluid Dynamics, eds. Buchler, J.R. and Gottesman, S.T., Ann NY Acad Sci, 617, p.17.Google Scholar
Buchler, J.R. and Goupil, M.J.: 1984, Astrophysical Journal 279, 394.Google Scholar
Buchler, J.R. and Goupil, M.J.: 1988, Astronomy and Astrophysics 190, 137.Google Scholar
Buchler, J.R., Goupil, M.J. and Kovács, G.: 1987, Phys. Letters A126, 177.Google Scholar
Buchler, J.R. and Kovács, G.: 1986a, Astrophysical Journal 303, 749.Google Scholar
Buchler, J.R. and Kovács, G.: 1986b, Astrophysical Journal 308, 661.Google Scholar
Buchler, J.R. and Kovács, G.: 1987a, Astrophysical Journal 318, 232.Google Scholar
Buchler, J.R. and Kovács, G.: 1987b, Astrophysical Journal, Letters to the Editor 320, L57.Google Scholar
Buchler, J.R. and Kovács, G.: 1992, Physica D. submitted.Google Scholar
Buchler, J.R., Kovács, G. and Goupil, M.J.: 1992, Astronomy and Astrophysics submitted.Google Scholar
Buchler, J.R., Moskalik, P. and Kovács, G.: 1990, Astrophysical Journal 351, 617.Google Scholar
Buchler, J.R. and Moskalik, P.: 1992, Astrophysical Journal 391, (in press. May).Google Scholar
Buchler, J.R. and Perdang, J., 1979, Astrophysical Journal 231, 524.CrossRefGoogle Scholar
Buchler, J.R. and Regev, O.: 1982, Astrophysical Journal 263, 312.Google Scholar
Buchler, J.R., Yueh, W.R., and Perdang, J.: 1977, Astrophysical Journal, 214. 510.Google Scholar
Canizzo, J.K., Goodings, D.A. and Mattei, J.: 1990, Astrophysical Journal, 357, L31.Google Scholar
Coullet, P. and Spiegel, E.A.: 1984, SIAM J. Appl. Math.. 43, 776.Google Scholar
Cvitanovich, P.: 1984, Universality in Chaos, (Bristol: Adam Hilger).Google Scholar
Dziembowski, W.: 1980, in Nonradial and Nonlinear Stellar Pulsations, eds. Hill, H.A. and Dziembowski, W.A. (Springer), p. 22.Google Scholar
Dziembowski, W.: 1982, Acta Astr. 32, 148.Google Scholar
Dziembowski, W. and Kovács, G.: 1984, Monthly Notices of the RAS 206, 497.Google Scholar
Dziembowski, W. and Krolikowska, M.: 1985, Acta Astr. 35. 5.Google Scholar
Dziembowski, W., Krolikowska, M. and Kosovitchev, A.: 1988, Acta Astr. 38, 61.Google Scholar
Glasner, A. and Buchler, J.R.: 1990, in The Numerical Modelling of Nonlinear Stellar Pulsations; Problems and Prospects, ed. Buchler, J.R., NATO ASI Ser. C302 (Kluwer, Dordrecht), p. 109.Google Scholar
Guckenheimer, J. and Holmes, P.: 1983, Nonlinear Oscillations, Dynamical Systems and Bifurcation Theory, (Springer, NY).Google Scholar
Goupil, M.-J., Auvergne, M. and Baglin, A.: 1988, Astronomy and Astrophysics, 196, L13.Google Scholar
Klapp, J., Goupil, M.-J. and Buchler, J.R.: 1985, Astrophysical Journal, 296, 514.Google Scholar
Kollath, Z.: 1990, Monthly Notices of the RAS 247, 377.Google Scholar
Kovács, G. and Kolláth, Z: 1988, in Multimode Stellar Pulsations, eds. Kovács, G. et al. (Kultura: Budapest), p. 33.Google Scholar
Kovács, G. and Buchler, J.R.: 1988a, Astrophysical Journal 324, 1026.Google Scholar
Kovács, G. and Buchler, J.R.: 1988b, Astrophysical Journal 334, 971.CrossRefGoogle Scholar
Kovács, G. and Buchler, J.R.: 1989, Astrophysical Journal 346, 898.Google Scholar
Kovács, G., Kisvarsányi, E. and Buchler, J.R.: 1990, Astrophysical Journal 351, 606.CrossRefGoogle Scholar
Livio, M. and Regev, O.: 1984, Astronomy and Astrophysics 148, 133.Google Scholar
Lloyd, , Evans, T.: 1971, Observatory 91, 159.Google Scholar
Moore, D. and Spiegel, E.A.: 1966, Astrophysical Journal 143, 871.Google Scholar
Moskalik, P.: 1985, Acta Astr. 35, 229.Google Scholar
Moskalik, P.: 1986, Acta Astr. 36, 333.Google Scholar
Moskalik, P. and Buchler, J.R.: 1989, Astrophysical Journal 341, 997.Google Scholar
Moskalik, P. and Buchler, J.R.: 1990, Astrophysical Journal 355, 590.Google Scholar
Moskalik, P. and Buchler, J.R.: 1991, Astrophysical Journal 366, 300.Google Scholar
Moss, F. and McClintock, P.V.E.: 1989, Theory of Continuous Fokker-Planck Systems, Vols. I and II, Cambridge University Press, (Cambridge, UK).Google Scholar
Papaloizou, J.C.B.: 1973, Monthly Notices of the RAS 162, 143; ibid. 162, 169.Google Scholar
Perdang, J.: 1983, Solar Physics 82, 297.CrossRefGoogle Scholar
Regev, O. and Buchler, J.R.: 1981, Astrophysical Journal 250, 776.Google Scholar
Saitou, M., Takeuti, M. and Tanaka, Y.: 1989, Publications of the ASJ 41, 297.Google Scholar
Serre, T. and Buchler, J.R.: 1992, Astronomy and Astrophysics submitted.Google Scholar
Simon, N.R.: 1982, Astrophysical Journal, Letters to the Editor 260, L87.Google Scholar
Simon, N.R.: 1986, Astrophysical Journal 311, 305.Google Scholar
Simon, N.R. and Lee, A.S.: 1981, Astrophysical Journal 248. 291.Google Scholar
Simon, N.R. and Moffett, T.J.: 1985, Publications of the ASP 97, 1078.Google Scholar
Simon, N.R. and Schmidt, E.G, 1976, Astrophysical Journal 205, 162.Google Scholar
Stratonovich, R.L.: 1965, Topics in the Theory of Random Noise, Vols. I and II, (New York: Gordon and Breach).Google Scholar
Takeuti, M.: 1985, Astrophysics and Space Science 109, 99.Google Scholar
Takeuti, M.: 1990, in The Numerical Modelling of Nonlinear Stellar Pulsations; Problems and Prospects, ed. Buchler, J.R., NATO ASI Ser. C302 (Kluwer, Dordrecht), p. 121.Google Scholar
Takeuti, M. and Aikawa, T.: 1980, Monthly Notices of the RAS 192, 697.Google Scholar
Takeuti, M. and Aikawa, T.: 1981, Sci Rep Tôhoku Univ. Eighth Ser. 2, 106.Google Scholar
Tanaka, Y. and Takeuti, M.: 1988, Astrophysics and Space Science 148, 229.Google Scholar
Vandakurov, Yu. V.: 1979, Sov Astr 23, 421.Google Scholar
Wallerstein, G.: 1958, Astrophysical Journal 127, 583.Google Scholar
Waltjer, J.: 1936, Bull. Atr. Inst. Netherlands 282, 17; 1937, ibid. 303, 193; 1946, ibid. 370, 125.Google Scholar