A historical sketch of the nonlinear theory of variable stars is outlined briefly. The main break-through came from the hydrodynamic study of stellar pulsation. From a theoretical point of view, coupling needs to be discussed more carefully. The impact of new opacities on the astrophysical problems is also discussed.

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.

Techniques for deriving amplitude equations for stellar pulsation are outlined. For the simplest such equations with multiple instabilities, the derivation of a map for the patterns of pulsation phases is described. This map gives the time between two successive maxima of pulsation in terms of the time between the previous pair, under suitable conditions. The phase differences can be regular, chaotic or hyperchaotic.

Tentatives to detect chaos in variable stars are presented. Constraints on the accuracy on individual measurements and length of the records, necessary to obtain defenitive conclusions, are discussed. Some directions for future improvements are sketched.

In this paper I will give overview of the stellar pulsation theory. Starting with basic equations I will discuss modal properties of oscillations and excitation mechanisms. I also mention briefly the effects of rotation.

Notwithstanding a great number of hypotheses, suggested for explaining superpositions of the light- and of the velocity variations of the ß Canis Majoris stars, no one of these does it satisfactorily. Possibly it is due to an inadequate elaboration of the non-linearly oscillation theory. Analysis and critical evaluation of the existing hypotheses are given by Mel’nikov and Popov (1970). Our explanation consists in existence of close frequencies corresponding to various oscillation modes which are non-linearly interacting.

Equations of motion of an ideal incompressible fluid under condition of preserving the equilibrium figure symmetry with respect to the equatorial plane (lateral oscillations) have the form (Baranov 1988):

One zone modeling of the irregular variability of red super-giants is intended with regard to the nonlinear coupling of finite amplitude pulsation with convection. The nonlocal mixing length is employed for the evaluation of the convective flux, the turbulent pressure and the turbulent power of temperature fluctuations. The radial pulsation and the Boussinesq convection are assumed for simplicity. The one zone is defined as the layer having the entropy maximum and the minimum at the bottom and at the top, respectively. The quasi-adiabatic approximation is consistent with this definition in fixing the zone to the same mass range. The spatial derivatives are evaluated under the assumption of homologous changes with the equilibrium homologous parameters. Then, a set of 6 simultaneous first order nonlinear ordinary differential equations are obtained as the one zone representation of the irregular variability of the convective envelope.

Basic properties of accretion disk instabilities are summarized. We first explain the standard disk model by Shakura and Sunyaev. In this model, the dominant sources of viscosity are assumed to be chaotic magnetic fields and turbulence in gas flow, and the magnitude of viscosity is prescribed by so-called a model. It is then possible to build a particular disk model. In the framework of the standard model, accretion disks are stationary, but when some of the basic assumptions are relaxed, various kinds of instabilities appear. In particular, we focus on the thermal limit-cycle instability caused by partial ionization of hydrogen (and helium). We demonstrate that the disk instability model well accounts for the basic observed features of outbursts of dwarf novae and X-ray nova. We then introduce other kinds of instabilities based on the α viscosity model. They are suspected to produce time variabilities observed on a wide range of timescales in close binaries and active galactic nuclei.

Motion of a charged particle around a black hole immersed in magnetic field is calculated. It is shown that this motion has a chaotic property depending on initial parameters.

The paper is an investigation of the magnetohydrodynamics (MHD) of the quasi-radial accretion onto a rotating black hole. It is demonstrated that the nonlinear effects in the accretion flow accelerate the rate of expansion of the Alfven mode. The accretion flow becomes unstable at the weak strength of magnetic field which is about one-thirtieth of the expected value by the linear theory.

Simplified stellar models like as one-zone models can be oscillated in chaotic state (Saitou et al. 1989). It is natural to consider that the irregular variations of Galactic X-ray sources might be caused from nonlinear oscillation of accretion disks or related flow. We studied the variability of Cen X-3 for applying the nonlinear theory to it.

The X-ray of Cen X-3 observed with Ginga from 22nd to 24th of March, 1989 are analyzed. Since Large Area Counters (LAC) of which the area of detector is 4,000 cm2 are used. The data used here is obtained the range of 1,000 sec from 14 h 25 m on March 23rd with the time resolution of 0.25 sec. The energy bands are 1.7~4.6 keV, 4.6~9.3 keV, 9.3~19 keV, and 19~37 keV, respectively.

Takeshima et al. (1991) show that Cen X-3 is at the high state, when the X-ray from the source is not veiled by non-ionized matter (Schreier et al. 1976). So that we ignore the effect of circumstellar matter around the source.

Population dynamics of multi-phased interstellar medium (ISM) is investigated by using the lattice model in position-fixed reaction. Interactions between three distinct phases of gas, cold clouds, warm gas, and hot gas give rise to cyclic phase changes in ISM. Such local phase changes are propagated in space, and stochastic steady-state spatial pattern is finally achieved. We obtain the following two characteristic patterns:

1. (1) When the sweeping rate of a warm gas into a cold component is relatively high, cold clouds associated with warm gas form small-scale clumps and are dispersively distributed, whereas hot gas covers large fraction of space.

2. (2) When the sweeping rate is relatively low, in contrast, warm gas and cold clouds are diffusively and equally distributed, while hot gas component is substantially localized.

Recent observations of several types of supergiant variable stars are reviewed: massive blue, yellow and red supergiants; classical and population II Cepheids; RV Tauri stars; yellow semi-regular (SRd) variables, including UU Herculis stars; and R Coronae Borealis stars. The emphasis is on non-linear aspects such as: amplitude and shape of the light and velocity curves; multiperiodicity, irregularity and chaos; long-term changes in period and amplitude; episodic and continuous mass loss.

The American Association of Variable Star Observers (AAVSO) is the largest organization of variable star observers in the world, with members in 42 countries. The purpose of the AAVSO is to coordinate variable star observing, done primarily by amateur astronomers, evaluate the accuracy of these observations, compile, process and publish them, and make them available to researchers and educators around the world. Over 6.5 million observations of variable stars have been complied since the AAVSO was founded in 1911. Those since about 1960 are in computerized form, and it is intended to have all observations in this form within the next year. About 250,000 observations are submitted to and archived by the AAVSO each year, over half of them from outside the USA. Most of the observations are visual, but there is also an active photoelectric program which concentrates on semi-regular and irregular stars such as small-amplitude red variables. In 1990, over 200 requests for AAVSO data and services were received from researchers and educators; this number has increased by a factor of 10 in the last two decades.

Visual observations of long period variable stars over 20 years were provided by the American Association of Variable Stars Observers, and were analysed as part of the preparation of the HIPPARCOS mission.

A set of frequencies is extracted from the light curve by using Fourier transform, preliminary Van Cittert deconvolution and comparison of the results obtained through different kinds of spectral windows. The same procedure is applied to the residual obtained after nonlinear fit of the main frequency. After final comparison of both sets, a nonlinear fit of the common frequencies gives the ‘clean’ power spectrum.

Different equilibrium stellar model (i) give theoretical linear nonadiabatic pulsation modes (vj) with their growth rates (ηj)i) (Tuchman 1978). The metallicity is taken between 0.005 and 0.02; the mixing length is λ = 1±0.2; the upper bound is r = 0.7. Assuming two peaks of the power spectrum to be the fundamental (vo) and first overtone (v1) modes, one looks for the corresponding models. The best one is selected by checking the other theoretical overtones they give. So are obtained the mass, the luminosity, the effective temperature and the effective radius of each star.

The existence of some variable stars producing very complicated light curves is well known. Theoretical calculations suggest that the irregular behaviour of the pulsation models in the RV Tauri and W Virginis regime is low dimensional is the result of period-doubling or tangent bifurcations (see e.g. Buchler and Kovács 1987, Kovács and Buchler 1988, Tanaka and Takeuti 1988).

The light variation of the RV Tauri star R Scuti covering 150 years was analyzed by Kolláth (1990). A striking similarity was found between the reconstructed attractor of the Rössler model and that of the light variation of R Scuti. This result confirms the theoretical prediction of the existence of chaos, but a discrepancy still exists between the theory and observation. We could not find evidence for low dimension (D = 2– 3). The analyses of other stars also show rather erratic behaviour (e.g. Cannizzo and Goodings 1988, Cannizzo et al. 1990). A possible answer for this discrepancy is the treatment of stochastic perturbations by convection (Perdang 1991).

I present a new method to test determinism, in particular the nonlinear behavior, in observed time series of pulsating stars, based on a recent prediction method which exploits the dynamical system theory. A method for filling gaps in data has thereby been constructed. Estimated bounds to the necessary embedding dimension can be obtained and chaotic divergences can be estimated.

Visual light curves of some Mira and semiregular variable stars are analyzed to search for their chaotic and statistical behaviour. Discussion are also given.