The project of an Optical Monitor (OM) for X-ray satellites, in particular the JET-X (Joint European Telescope for X-ray astronomy) experiment (Wells et al., 1991), derives from the scientific need of having complete data coverage at various wavelengths, UV and optical, of the observed X-ray sources, because these data are essential for a deeper understanding of the various classes of objects. When studying variable sources and/or transient astronomical phenomena, one needs that the multifrequency observations be performed essentially at the same time, because it is the knowledge of the simultaneous optical and X-ray behaviour of a source which contributes substantially to the clarification of its nature. In principle optical observations simultaneous with X-ray ones can be performed from ground based telescopes. However the complexity of satisfying the constraints typical of the optical telescopes (weather conditions, source observability) and of the X-ray instrumentation (e.g. orbital constraints) lead inevitably to a substantial loss of observing time. Therefore the only practical way of having an optimal utilization of the time available for X-ray observations, together with the wealth of scientific potential of simultaneous UV-optical observations, is to have a small telescope to be part of the same space mission.

General properties of Luminous Blue Variables (LBV) have been reviewed by Lamers (1987). The LBV’s are all close to the Humphreys-Davidson luminosity upper limit. The semi periods of the photometric microvariations with ΔV ~ 0.1m are about twice as large as for normal supergiants of the same L and Teff, and 4-20 times larger than the fundamental mode for radial pulsations. So, it is likely due to nonradial pulsations. During shell ejections, which accompanied by moderate photometric variations of ΔV ~ 1m, Lbol remains constant. This fact has been explained by the quasiperiodic variations of Teff and the radius of pseudo-photosphere (Lamer 1986). LBV’s are less stable than normal supergiants so that any internal instability has more effect on their envelopes. The nature of these instabilities remains unknown. We have found some interesting peculiarities which can throw light to this problem.

The pulsation motions of the limit-cycle model can be described as a superposition of the Fourier harmonics, in the adiabatic layers each harmonics being identified with the corresponding standing wave. Near the resonance Π0/Πl = k the harmonics of order k is also identified with the overtone of order l The spectra of the oscillatory moment of inertia obey to the power dependence on the Fourier harmonics order k. In cepheids with periods shorter than 9 days the bump is due to the wave packet generated by the second overtone, whereas at periods longer than 10 days the bump feature is due to the traveling pulse reflected off the stellar core.

Phenomena of bifurcation in hydrodynamic stellar models of radial pulsation are reviewed. By changing control parameters of models, we can see qualitatively different pulsation behaviors in hydrodynamic models with transitions due to various types of bifurcation.

In weakly dissipative models (classical Cepheids). the bifurcation is induced by modal resonances. Two types of the modal resonances found in models are discussed: The higher-harmonic resonances of the second overtone mode in the fundamental mode pulsator and of the fourth overtone mode in the first overtone pulsator are relevant to observations. The subharmonic resonance between the fundamental and first overtone modes is confirmed in classical Cepheid models.

In strongly dissipative models (less-massive supergiant stars), the bifurcation of nonlinear pulsation is induced by the hydrodynamics of ionization zones as well as modal resonances. The sequence of the bifurcation sometimes leads to chaotic behaviors in nonlinear pulsation. The transition routes from regular to the chaotic pulsations found in models are discussed with respect to the theory of chaos in simple dynamical systems: The cascade of period-doubling bifurcation is confirmed to cause chaotic pulsation in W Virginis models. For models of higher luminosity, the tangent bifurcation is found to lead intermittent chaos.

Finally, hydrodynamic models for chaotic pulsation with small amplitudes observed in the post-AGB stars are briefly discussed.

The current observational and theoretical status of the double-mode variables is reviewed. Focusing mostly on the RR Lyrae stars, we address the question of the observational evidence of modal stability. The problem of stationarity is a crucial issue in the modelling of these stars.

We mention past efforts in hydrodynamical and analytical modelling together with a detailed discussion of some very recent results. It is suggested that stochastic forcing due to turbulent convection may play a crucial role in exciting some marginally stable modes in the limiting pulsation. The latest hydrodynamical results first demonstrate that purely radiative models are able to show permanent double-mode behavior in the relevant period regime of RRd stars. The reason for the previous lack of double-mode behavior is attributed to the large dissipation, i.e. artificial viscosity, generally used in the codes to ensure numerical stability and to obtain amplitudes comparable to the observed ones.

We think that better models should include some physical dissipation, most probably turbulent convection, and a more accurate numerical treatment of the radiative hydrodynamics.

We have performed recently a survey of the nonlinear hydrodynamical models of the BL Her-type variables (Buchler & Moskalik 1992). Within this project we have studied several sequences of models, i.e., families in which onlyTeff has been varied from model to model, while all other stellar parameters have been kept constant. The fundamental mode pulsations of each model have been converged to strict periodicity with the relaxation code (Stellingwerf 1974). Such approach speeds up the calculations and simultaneously yields information about the stability properties of the resulting limit cycles. In all studied sequences except one, we have found a narrow-range of Teff (typically 100-150K), in which regular solution becomes unstable towards a period doubling bifurcation. The instability has its origin in a half-integer resonance, namely the 3:2 coupling between the fundamental mode and the first overtone (cf. Moskalik & Buchler 1990; hereafter MB90). This is the same resonance, which also causes period doubling in the models of classical Cepheids (Moskalik & Buchler 1991). The bifurcation leads to stable period-two oscillations, characterized by an RV Tau-like, albeit strictly periodic behavior of all variables. In other words, the pulsation light curves and velocity curves will exhibit two alternating minima (as well as maxima) of different values.

A. N. Cox recently showed that a 20% opacity decrease in the 20,000–30,000 K region as indicated by the new Livermore OPAL opacities reconciles the discrepancy between pulsation and evolution masses of double-mode RR Lyrae variables. Nonlinear hydrodynamic calculations were performed for RR Lyrae models of mass 0.75M⊙, 51L⊙, and Z = 0.0001 (Osterhoff II type) including this opacity decrease. The Stellingwerf periodic relaxation method was used to converge the models to a limit cycle, and the Floquet matrix eigenvalues calculated to search for a tendency of the fundamental mode to grow from the full-amplitude overtone solution, and the overtone mode to grow from the full-amplitude fundamental solution, thereby predicting double-mode behavior. Models of Teff < 7000 K with the opacity decrease have positive fundamental-mode growth rates in the overtone solution, in contrast to earlier results by Hodson and Cox (1982), and models with Teff > 7000 have positive 1st overtone growth rates in the fundamental-mode solution, but double-mode behavior was not found.

Both linear and nonlinear calculations of the 331 day, long period variable star Mira have been undertaken to see what radial pulsation mode is naturally selected. Models are similar to those considered in the linear nonadiabatic stellar pulsation study of Ostlie and Cox (1986). Models are considered with masses near one solar mass, luminosities between 4000 and 5000 solar luminosities, and effective temperatures of approximately 3000 K. These models have fundamental mode periods that closely match the pulsation period of Mira. The equation of state for the stellar material is given by the Stellingwerf (1975ab) procedure, and the opacity is obtained from a fit by Cahn that matches the low temperature molecular absorption data for the Population I Ross-Aller 1 mixture calculated from the Los Alamos Astrophysical Opacity Library. For the linear study, the Cox, Brownlee, and Eilers (1966) approximation is used for the linear theory variation of the convection luminosity. For the nonlinear work, the method described by Ostlie (1990) and Cox (1990) is followed. Results showing internal details of the radial fundamental and first overtone modes behavior in linear theory are presented. Preliminary radial fundamental mode nonlinear calculations are discussed. The very tentative conclusion is that neither the fundamental or first overtone mode is excluded from being the actual observed one.

Walraven, Walraven and Balona recently discovered several new periodicities in addition to the well-known fundamental and first overtone periods of the high-amplitude δ Scuti star AI Velorum. Linear nonadiabatic pulsation calculations were performed for an AI Velorum model of mass 1.96M⊙, 24.05L⊙, and Teff 7566 K for the radial and low-degree nonradial modes to help verify the tentative identifications made by Walraven, et al. Comparison of the calculated periods with the observations suggests some alternatives to the identifications proposed by Walraven, et al.

With the advent of new astrophysical opacities it seems appropriate to discuss the need for a full radiative transfer (RT) theory instead of the usual equilibrium diffusion theory used in most nonlinear pulsation codes. Early studies on the importance of RT in the calculation of light curves for Cepheid models showed little effect over diffusion theory. The new opacities though may help to explain the “bump” mass discrepancy problem. For RR Lyrae models the use of RT theory causes some effects both in the color differences (U-B) as well as in the light curves. New opacities help to explain the period ratios for double mode RR Lyrae and beat Cepheids. A new area of research is in the modeling of stars with high luminosity to mass ratios that show tendencies for doubling and transitions to chaos, such as W Virginis and RV Tauri stars. For these stars it has been shown that RT is necessary in calculating their light curves and that the understanding of the shock dynamics depends on the transfer of lines in the pulsating RT dependent atmospheres (Fokin 1991).

The atmospheres of classical Cepheids cannot be represented adequately by a sequence of hydrostatic equilibrium atmospheric models (see Sasselov and Lester 1990 and references therein). Observational evidence for shocks is also available at least since Kraft (1956). Both the disequilibrium and the non-linear phenomena affect the emergent spectra and spectral line profiles of the Cepheids. In the lower amplitude variables the effects may be subtle, yet still lead to significant systematic discrepancies.

We have developed the code HERMES for time-dependent treatment of hydrodynamics and non-LTE radiative transfer in the atmospheres of classical Cepheids with periods shorter than 12 days. Our approach is applicable to stars in which the formed shock waves do not dominate the energy balance of the atmosphere. Consecutive detailed calculations are performed for several multi-level model atoms, including H, Ca II, Mg II, He I and II. We use a 1-D explicit conservative upwind monotonic second-order (in time and space) Godunov-type Lagrangian hydrodynamic scheme. Being a characteristics based scheme it allows natural handling of boundary conditions. The scheme is stable and without artificial dissipation, a crucial necessity for physically meaningful radiative transfer solutions (see Roe 1990, Sasselov and Raga 1991 for more details).

Some results of linear adiabatic and nonlinear pulsation models of first overtone mode Cepheids are discussed. New and augmented opacities and a nonstandard mass-luminosity relation have been taken into account. The models indicate the possible importance of the resonance P1/P4 = 2 near P1 = 3 days. The resonance could explain the observed characteristics of the light curve shape of first overtone mode Cepheids.

Modal coupling oscillation models for the stellar radial pulsation and coupled-oscillators are reviewed. Coupled-oscillators with the second-order and third-order terms seemed to behave non-systematically. Using the equation by Schwarzschild and Savedoff (1949) with the dissipation term of van del Pol’s type which is third-order, we demonstrate the effect of each term. The effects can be understood by the terms of the nonlinear dynamics, which is recently developing, that is. phase-locking, quasi-periodicity, period doubling, and chaos. As the problem of stellar pulsation, especially of double-mode cepheids on the period-ratio, we examine the dependence on the stellar structure from which the coupling constants in the second-order terms are derived. Eigen functions for adiabatic pulsations had been used for the calculation of the constants. It is noted that only two set of the constants are available, that is, for the polytrope model with n = 3 and a cepheid model without convection. Some examples of nonlinear dynamical effects will be shown.

It is shown that if the constants were suitable values, the period-ratio of double-mode cepheids is probably realized. The possibility is briefly suggested.

The dependence of coupling constants in a coupled oscillator model is examined with simplified methods. The Lyapunov exponents are preliminary introduced for the model. The behaviors of oscillator model are examined in a parameter plane. So-called the Arnold’s tongues for phase-locking states are observed in fractal patterns.

The second order theory of coupling is discussed regarding the radial pulsation of stellar models which are constructed ignoring convection. The formula including the nonadiabatic effect is presented. Numerical values given for model classical cepheids are considerably greater than the adiabatic values.

The third order theory of coupling is discussed regarding the radial pulsation of stellar models.

The effect of convection on the strength of coupling is examined. It is found that in Cepheid models the inclusion of convection smooths the sharp peak of entropy perturbation in the ionization region and reduces significantly the coupling.

We study the convective nonlinear one-zone models of the pulsating variable stars. In the small convective case, the solution shows chaotic behavior through the period doubling bifurcation as the temperature is lower although the temperature at which the chaos appears is lower than in the case of no convection. On the other hand, in the strong convective case, the solution only shows period-one limit cycle.

A unified model for outbursts of dwarf novae is proposed based on the disk instability model in cataclysmic variable stars. In this model, two different intrinsic instabilities (i.e., the thermal instability and the tidal instability) within accretion disks are considered in non-magnetic cataclysmic variable stars. It is suggested that all of three sub-classes of dwarf novae (i.e., U Gem-type, Z Cam-type and SU UMa-type dwarf novae) may be explained in terms of two model parameters of the orbital period of the binary and of the mass transfer rate within the framework of the disk instability model.

Linear analysis shows that radial oscillations in accretion disks around compact object are overstable to axisymmetric perturbation under a variety of conditions. Furthermore, numerical simulations confirm that the radial oscillations induce quasi-periodic modulations of the disk luminosity. The disk oscillation model may be responsible for quasi-periodic oscillations (QPOs) observed in low mass X-ray binaries (LMXBs), cataclysmic variables (CVs), and other compact objects.