The AAVSO lightcurve data of the irregularly pul- sating star AC Herculis of the RV Tau class are analysed. The lightcurveisshownnottobecompatiblewithaperiodic,oreven multiperiodicpulsation,evenifallowanceismadeforevolution. Instead the best explanation is that the irregularly alternating cycles are a manifestation of low dimensional chaos. Our flow reconstruction suggests that the dynamics can be successfully embedded in 3 dimensions. Further, the (Lyapunov) fractal di- mension of the underlying dynamic attractor is computed to be 2:05 < dL < 2:45, independent of the dimension of the recon- struction space. It is smaller than the value of dL 3.1 found for the more irregular lightcurve of R Sct. From these results we are led to conclude that the lightcurve is likely to have been generated by a 3 dimensional dynamics, i.e. by 3 rst order ODEs. In simpler language, it expresses the remarkable fact that the observational value at any time s(tn) is a unique func- tion h of the values at only three preceding observations, viz. s(tn+1 )= h s ( t n );s(tn 1);s(tn 2) . In this paper we analyze the AAVSO data set of the lightcurve of AC Herculis that shows irregular pulsations with large cycle to cycle alternations in the lightcurve. The paper is organized in parallel to the analysis of R Scuti (Buchler, Serre, Kollath & Mattei 1995 hereafter BSKM, Buchler, Kollath, Serre & Mattei 1996, hereafter BKSM). Despite their membership in the same RV Tau class the lightcurves of these two stars are quite dif- ferent. AC Her has a cycle time of 35 days, about half that of R Sct, has a much lower pulsation amplitude, and is much less irregular. From a practical point of view, the available AAVSO data set is shorter, there are more gaps and fewer points per cy- cle, and the relative noise level is higher, all of which make the analysis somewhat harder. In Sect. 2 we show on the basis of Fourier spectral analysis and theoretical considerations that the pulsations are not com- patible with periodic or multiperiodic behavior, even when evo- lution is allowed for. The pulsations must therefore be either of a chaotic or of a stochastic nature. (We follow the general trend of distinguishing the two by calling chaotic an erratic signal that is generated by a deterministic low dimensional dynamics whose form can be found, at least in principle, and stochastic a very high dimensional dynamics that is too complicated to be characterized otherwise than stochastically.) In Sect. 3 we perform a nonlinear analysis that shows that the observational data are indeed compatible with low dimensional chaos, and we determine its quantitative properties. This is followed by a discussion in Sect. 4, and we conclude in Sect. 5.

We describe a powerful, recently developed method for nonlinear time-series analysis. The method allows one to check whether the given signal has a dominant component that been generated by a low-dimensional nonlinear dynamics. Furthermore it allows one to extract properties of this dynamics from the mere knowledge of the given scalar (single) observed quantity. In the context of variable stars this is normally the luminosity of the star, or possibly its radial velocity. The method is tailored to irregular signals and thus complements the classical techniques of analysis which apply only to multi-periodic signals. The ultimate purpose is to develop a better physical understanding of the pulsations and to derive novel astrophysical constraints from irregular light curves.Before applying the global flow reconstruction to signals of unknown properties it is imperative to test it on a well known system. For that purpose we have applied it to the well studied Rossler oscillator which, we note, has a behavior that is similar to the one encountered in the numerical model pulsations of W Virginis stars. For the analysis we allow ourselves only a short section of the temporal behavior of only one of the 3 Rossler variables to infer properties of the whole attractor. In order to make the test more realistic, Gaussian noise has also been added to the Rossler oscillator data. The method is shown to perform very well, producing synthetic signals that, when chaotic, are very close to the original one, with similar Fourier spectra. The map that is obtained from the data allows one then to quantify the complexity of a chaotic signal, e.g. with the help of Lyapunov exponents and fractal dimensions of the synthetic signals. These quantities appear to be fairly robust.The minimum embedding dimension for the reconstruction with the first Rossler variable is found to be 3. Importantly, also, even though only one Rossler variable is assumed to be known the method recovers the 'physical' dimension 3 of the RGssler band. The method works well even when large amounts of noise are added to the signal prior to the analysis.In a twin paper we analyze the pulsations of a W Virginis model. Applications to observational data of R Set and of AC Her are presented in companion papers.

It is first shown that the observational light curve data of R Scuti, a star of the RV Tau type, is not multi-periodic, and that it cannot have been generated by a linear stochastic (AR) process. By default, the signal must be a manifestation of deterministic chaos. We use a novel nonlinear time-series analysis, the global flow reconstruction technique, to probe the properties of the irregular pulsation cycles. We show in particular that the chaotic dynamics of this star's complicated lightcurve is captured by a simple 4D polynomial map or flow (4 first order ODEs). Importantly also, the method allows us to quantify an irregular signal which has the potential benefit for extracting novel stellar constraints from an irregular light-curve. Finally, from the low dimensionality 4 of the flow we can infer a simple physical picture of the pulsations, and arguments are presented that the pulsations of R Sct are the result of the nonlinear interaction of two vibrational normal modes of the star.

Observational data of the light curve of the variable star R Scuti are subjected to a powerful recently developed nonlinear dynamics approach. This is the first time it is shown that the irregular pulsations of a star are described by a chaotic dynamics with an embedding dimension of 4. The results are relatively robust with respect to the presmoothing of the observational data, as well as to the parameters of the method (such as the delay and sampling rate). This low dimension suggests that the complex pulsational behavior is a result of the nonlinear interactions between just two vibrational modes.

Annals of the New York Academy of SciencesVolume 773, Issue 1 p. 1-13 Chaos in Observational Variable Star Data—Irregularity from the Nonlinear Interaction of Standing Waves?† J. ROBERT BUCHLER, J. ROBERT BUCHLER Physics Department University of Florida Gainesville, Dlorida 32611Search for more papers by this authorZ. KOLLÁTH, Z. KOLLÁTH Physics Department University of Florida Gainesville, Dlorida 32611Search for more papers by this authorTHIERRY SERRE, THIERRY SERRE Physics Department University of Florida Gainesville, Dlorida 32611Search for more papers by this author J. ROBERT BUCHLER, J. ROBERT BUCHLER Physics Department University of Florida Gainesville, Dlorida 32611Search for more papers by this authorZ. KOLLÁTH, Z. KOLLÁTH Physics Department University of Florida Gainesville, Dlorida 32611Search for more papers by this authorTHIERRY SERRE, THIERRY SERRE Physics Department University of Florida Gainesville, Dlorida 32611Search for more papers by this author First published: December 1995 https://doi.org/10.1111/j.1749-6632.1995.tb12157.xCitations: 5 † This research was supported in part by National Science Foundation Grant AST92–18068, in part by an RDA grant at the University of Florida, in part by the French Ministére pour la Recherche et l'Espace, and in part by an RCI grant from IBM through the University of Florida. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat REFERENCES 1 Cox, J. P. 1980. Theory of Stellar Pulsation. Princeton Univ. Press. Princeton , N.J. 2 Wheatly, C. W., G. W. Swift & A. Migliori. 1986. Los Alamos Sci. 14: 2. 3 Ludendorff, H. 1928. Handb. Astrophys. 6: 49. 4 Christy, R. F. 1972. Astrophys. J. 172: 419. 5 Buchler, J. R. & G. Kovács. 1987. Astrophys. J. Lett. 320: L57–L62. 6 Kovács, G. & J. R. Buchler. 1988. Astrophys. J. 334: 971. 7 Berge, P., Y. Pomeau & C. Vidal. 1984. Order Within Chaos. Wiley. New York . 8 Ott, E. 1993. Chaos in Dynamical Systems. Cambridge Univ. Press. Cambridge , England . 9 Buchler, J. R. 1990. Ann. N.Y. Acad. Sci. 617: 17. 10 Duong-Van, M. 1987. Chaos 87. North-Holland. Amsterdam . 11 Constantin, P., C. Foias, B. Nicolaenko & R. Temam. 1989. Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Applied Mathematical Sciences, 70. Springer-Verlag. New York . 12 Sauer, T., J. A. Yorke & M. Casdagli. 1991. J. Stat. Phys. 65: 579. 13 Weigend, A. S. & N. A. Gerschenfeld. 1994. Time Series Prediction. Addison-Wesley. Reading , Mass . 14 Abarbanel, H. D. I., R. Brown, J. J. Sidorowich & L. S. Tsimring. 1993. Rev. Mod. Phys. 65: 1331. 15 Wisdom, J. 1987. Proc. R. Soc. London. A413: 109. 16 Ruelle, D. 1994. Phys. Today. July 24. 17 Buchler, J. R., T. Serre, Z. Kolláth & J. Mattei. 1995. Phys. Rev. Lett. 74: 842. 18 Serre, T., Z. Kolláth & J. R. Buchler. 1995. submitted for publication in Astron. Astrophys. 19 Buchler, J. R., Z. Kolláth, T. Serre & J. Mattel 1995. submitted for publication in Astrophys. J. 20 Kolláth, Z., J. R. Buchler, T. Serre & J. Mettei. 1996. in preparation. 21 Buchler, J. R. 1990. NATO ASI Series C302: 1. Kluwer. Dordrecht , The Netherlands . 22 Brown, R. 1993. UCSD-INLS. Preprint. 23 Mattei, J., M. Saladyga, E. O. Waagen & C. M. Jones. 1988. AAVSO Monograph 3. 24 Kollath, Z. 1990. Mon. Not. R. Astron. Soc. 247: 377. 25 Veldhuizen, T. & J. R. Percy. 1989. J. AAVSO 18: 97. 26 Saitou, M., M. Takeuti & Y. Tanaka. 1989. Astron. Soc. Japan 41: 297. 27 Gingold, R. A. 1976. Astrophys. J. 204: 116. 28 Brown, R., N. F. Rulkov & E. R. Tracy. 1994. Phys. Rev. E. 49: 3784. 29 Mindlin, G. B. & R. Gilmore. 1992. Physica D58: 229. 30 Letellier, C., P. Dutertre & F. Soufi. 1994. “ Charactérisation topologique d'une cephéide de type II modélisée par un code hydrodynamique. private communication. 31 Gillet, D., A. Duquennoy, P. Bouchet & C. Gouiffes. 1989. Astron. Astrophys. 215: 316. 32 Buchler, J. R. 1993. In Nonlinear Phenomena in Stellar Variability, M. Takeuti and J. R. Buchler, Eds. Kluwer. Dordrecht , The Netherlands . Reprinted from 1993. Astrophys. Space Sci. 210: 1. 33 Buchler, J. R. & N. E. G. Buchler. 1994. Astron. Astrophys. 285: 213. 34 Aikawa, T. 1990. Astrophys. Space Sci. 164: 295. 35 MOSKALIK, P. & J. R. Buchler. 1990. Astrophys. J. 355: 590. Citing Literature Volume773, Issue1Waves in AstrophysicsDecember 1995Pages 1-13 ReferencesRelatedInformation