Abstract
Simple models of nonlinear stellar pulsation, whose temporal behavior may reproduce some of the observed features of different classes of variable stars, have been studied. The temporal behavior of dynamical variables of these models exhibits a cascade of period doubling chaos, depending on the specific values of the various control parameters. A multifractal detrended fluctuation analysis (MFDFA) method is further used to identify the scaling behavior of such synthetic time series. The MFDFA of the considered time series, for various models of nonlinear stellar pulsation, shows that the observed multifractal nature is due to long-range correlations. The pulsating star with increased nonadiabaticity and the star with increased convective luminosity, as represented by the simulated data, is shown to possess a strange attractor with noninteger correlation dimension that lies between 2–3. Also the problem of synchronization in coupled nonlinear pulsation models has been investigated using permutation entropy—a complexity measure of the system. The computed order parameter, Γ, representing the correlation of computed permutation entropy for different segments of the simulated time series of displacement of two nonidentical oscillators, has been further used to find the critical coupling parameter for general synchronization of the oscillators.
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References
Auvergne, M., Baglin, A.: Astron. Astrophys. 168, 118 (1986)
Baker, N.H.: In: Stein, R.F., Cameron, A.G.W. (eds.) Stellar Evolution, p. 333. Plenum Press, New York (1966)
Badii, R.: Riv. Nuovo Cimento 12, 1 (1989)
Bandt, C., Pompe, B.: Phys. Rev. Lett. 88, 174102 (2002)
Boccaletti, S., Kurth, J., Osipov, G., Valladares, D.L., Zhou, C.S.: Phys. Rep. 366, 1 (2002)
Buchler, J.R., Kovacs, G.: Astrophys. J. 303, 749 (1986)
Buchler, J.R., Kovacs, G.: Astrophys. J. 320, L57 (1987)
Collet, P., Lebowitz, J., Porzio, A.: J. Stat. Phys. 47, 609 (1987)
Das, M.K., Bhattachrjee, A., Bhatraju, N.K., Yuasa, M., Saha, L.M.: Bull. Calcutta Math. Soc. 104, 561 (2012)
Dziembowski, W., Kovacs, G.: Mon. Not. R. Astron. Soc. 206, 497 (1984)
Farmer, J.D., Ott, E., Yorke, J.A.: Physica 7D, 153 (1983)
Feigenbaum, M.: J. Stat. Phys. 46, 5 (1987)
Glazier, J.A., Libchaber, A.: IEEE Trans. Circuits Syst. 35, 790 (1988)
Grassberger, P., Procaccia, I.: Physica 9D, 989 (1983)
Gukenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, Berlin (1993)
Goupil, M., Auvergne, M., Baglin, A.: Astron. Astrophys. 196, L13 (1988)
Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Schraiman, B.: Nucl. Phys. B, Proc. Suppl. 2, 513 (1986)
Hu, J., Gao, J., Wang, X.: J. Stat. Mech. 9, 02066 (2009)
Icke, V., Frank, A., Heske, A.: Astron. Astrophys. 258, 341 (1992)
Kantelhardt, J.W., Zchiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley, H.E.: Physica A 316, 87 (2002)
Kiss, L.L., Szatmary, K.: Astron. Astrophys. 390, 585 (2002)
Kolláth, Z.: Mon. Not. R. Astron. Soc. 247, 377 (1990)
Kolláth, Z., Buchler, J.R., Serre, T., Mattei, J.: Astron. Astrophys. 329, 147 (1998)
Liu, Z.: Europhys. Lett. 68, 19 (2004)
Liu, Z., Chen, S.: Phys. Rev. E 55, 6651 (1997)
Movahed, M.S., Jafari, G.R., Ghasemi, F., Rahavar, S., Tabar, M.: J. Stat. Mech. 6, 02003 (2006)
Munteanu, A., Bono, G., Jose, J., Garcia-Berro, E., Stellingwerf, R.F.: Astrophys. J. 627, 454 (2005)
Munteanu, A., Garcia-Berro, E., Jose, J.: Mon. Not. R. Astron. Soc. 341, 855 (2003)
Muzy, J.F., Bacry, R., Arneodo, A.: Int. J. Bifurc. Chaos 4, 245 (1994)
Packard, N., Crutchfield, J., Farmer, D., Shaw, R.: Phys. Rev. Lett. 45, 712 (1980)
Perdang, J.M.: In: Buchler, J.R., Perdang, J.M., Spiegel, E.A. (eds.) Chaos in Astrophysics, p. 11. Reidel, Dordrecht (1985)
Pyragas, K.: Phys. Rev. E 54, 4508 (1996)
Rössler, O.E.: Phys. Lett. 57A, 397 (1976)
Saitou, M., Takeuti, M., Tanaka, Y.: Publ. Astron. Soc. Pac. 41, 297 (1989)
Scafetta, N., Griffin, L., West, B.J.: Physica A 328, 561 (2003)
Staniek, M., Lehnertz, K.: Int. J. Bifurc. Chaos 17, 3729 (2007)
Stellingwerf, R.F.: Astrophys. J. 303, 119 (1986)
Takeuti, M., Tanaka, Y., Okazaki K., S.: Sci. Report Tohoku Univ., 8th Ser, vol. 9, p. 151 (1989)
Tanaka, Y.: Publ. Astron. Soc. J. 63, L49 (2011)
Tanaka, Y., Saha, L.M.: Publ. Astron. Soc. J. 64, L8 (2012)
Tanaka, Y., Takeuti, M.: Astrophys. Space Sci. 148, 229 (1988)
Tanaka, Y., Takeuti, M.: Astrophys. Space Sci. 180, 157 (1991)
Votruba, V., Koubsky, P., Korcakova, D., Hroch, F.: Astron. Astrophys. 496, 217 (2009)
Wang, W., Liu, Z., Hu, B.: Phys. Rev. E 62, 2114 (2000)
Acknowledgements
We thank the University of Delhi for providing the necessary facilities to carry out the work. (M.K.D.) was supported through a research grant from University of Delhi. One of us (N.K.B.) is thankful to the Department of Science & Technology (DST), India for financial support. We would like to further thank the anonymous reviewer for his encouraging suggestions, which led to a substantial improvement of the present work.
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Bhatraju, N.K., Das, M.K. On multifractality and synchronization of nonlinear stellar pulsators. Astrophys Space Sci 352, 83–94 (2014). https://doi.org/10.1007/s10509-014-1887-x
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DOI: https://doi.org/10.1007/s10509-014-1887-x