Chaos in the Stars
Irregular Variable stars
Lightcurve of the variable star R Scuti
(data from American Association of Variable Star Observers)
1. Time-series analysis of observational lightcurve of the star R Scuti
The nonlinear time-series analysis alows us to infer the properties
of the physical dynamics of the star from the mere knowledge of the observed
lightcurve.
Using a 'flow reconstruction' technique [1] we have constructed
a map that generates the observed lightcurve of R Scuti [2, 3].
The remarkable properties of this map are its low dimension of 4.
This means that a simple set of four order first order differential equations
can generate the complex lightcurve of R Sct.
The pulsations of this star are quite violent, with the luminosity
varying up to factors of 40, with
shockwaves and ionization fronts criss-crossing
the stellar envelope. The uncovered low dimensional behavior is thus perhaps
surprising.
As a byproduct of the time-series analysis we find that the fractal
dimension of the attractor is approximately 3.1 - 3.2 .
From the map we also infer the interesting physical result that the
complicated lightcurve is the result of the nonlinear, resonant
interaction between
two modes of pulsation, the first one linearly unstable and the second
one, linearly stable, with almost twice the frequency of the first.
An analysis of additional stars gives strong evidence that this resonant
interacion mechanism is generic, at least for a subgroup or irregularly
pulsating stars.
Buchler, J. R., Serre, T., Kollath, Z. & Mattei, J. 1995,
A Chaotic Pulsating Star -- The Case of R~Scuti, Physical Review
Letters 74, 842.
Buchler, J. R., Kollath, Z. & Cadmus, R. 2001, Chaos in the
Music of the Spheres, Proceedings of CHAOS 2001, July 22 - 26, Potsdam,
Germany, (in press);
astro-ph/0106329
Buchler, J.R., 1996, Search for Low-Dimensional Chaos in
Observational Data, International School of Physics "Enrico
Fermi", Course CXXXIII, "Past and Present Variability of the
Solar-Terrestrial System: Measurement, Data Analysis and Theoretical
Models", Eds. G. Cini Castagnoli & A. Provenzale, 275-288,
Società Italiana de Fisica, Bologna, Italy. chao-dyn/9707012
Serre, T., Kollath, Z. & Buchler, J. R., 1996, Search
For Low--Dimensional Nonlinear Behavior in Irregular Variable Stars --
The Global Flow Reconstruction Method, Astronomy & Astrophysics
311, 833.
Buchler, J. R., Kollath, Z., Serre, T. & Mattei, J. 1996,
A Nonlinear Analysis of the Variable Star R Scuti, Astrophysical
Journal, 462, 489.
astro-ph/9707116
Buchler, J.R. 1993, A Dynamical Systems Approach to Nonlinear
Stellar Pulsations, in Nonlinear Phenomena in Stellar Variability,
Eds. M. Takeuti & J.R. Buchler, Dordrecht: Kluwer Publishers, reprinted
from 1993, Ap&SS, 210, 1-31. paper
In French: Buchler, J. R. & Kollath, Z. 2001, Du chaos dans la
musique des
étoiles , (in french), Comptes Rendus du Colloque sur le
"Chaos temporel et chaos spatio-temporel", 24-25 sep. 2001, Le
Havre, France (in press); nlin.CD/0109028
postscript
file, with higher resolution figures
Buchler, J. R., Kolláth, Zoltan and Cadmus, R., 2004, Evidence
for Low-Dimensional Chaos in Semiregular Variable Stars, Astrophysical
Journal, Vol. 613, pp. 532 - 547.
http://xxx.lanl.gov/abs/astro-ph/0406109
 
ApJ   .ps.gz
2. Numerical Hydrodynamic simulations
State of the art hydrodynamical simulations of the nonlinear behavior
of similar types of stars (W Virginis type stars) that were performed a
decade ago [5, 6] had predicted the chaotic nature of these pulsations.
Indeed, sequences of models show a period doubling cascade as a
control parameter (the effective temperature of the equilibrium stellar
model) is varied. One-dimensional first return maps of the pulsation maxima
for an irregularly pulsating model clearly display a Feigenbaum quadratic
maximum.
The "flow reconstruction" technique indicates that the pulsations
take place in a 3D phase-space [7]. The fractal dimension of the attactor
is approximately 2.05-2.10. The attractor bears great similarity to the
well known Roessler system.
Template
for the attractor.
A topological analysis based on the way the orbits of the attractor
are organized spatially [8] shows that the attractor is actually only superficially
similar to the Roessler system.
Buchler, J.R. & Kovacs, G. 1987, Period-Doubling
Bifurcations and Chaos in W Virginis Models, Astrophysical
Journal Letters 320, L57-62
Kovacs G. & Buchler J.R. 1988, Regular and Irregular
Pulsations in Population II Cepheids , Astrophysical Journal 334,
971.
Serre, T., Kollath, Z. & Buchler, J. R., 1996,
Search For Low--Dimensional Nonlinear Behavior in Irregular Variable Stars
-- The Analysis of W Vir Model Pulsations, Astronomy &
Astrophysics, 311, 845.
Letellier, C., Gouesbet, G., Soufi, F., Buchler, J.R. & Kollath,
Z. 1996, Chaos In Variable Stars: Topological Analysis of W Vir
Model Pulsations, Chaos, 6, 466
Work supported by National Science Foundation.
Updated November 2005