ABSTRACT. Using the phase plane diagrams, the phase light curves of a group of the Mira-type stars and semi-regular variables are analyzed. As generalized coordinates х and ẋ, we have used т -the brightness of the star and its phase derivative. We have used mean phase light curves using observations of various authors. The data typically span a large time interval (nearly a century). They were compiled from the databases of AAVSO, AFOEV, VSOLJ, ASAS and approximated using a trigonometric polynomial of statistically optimal degree. As the resulting approximation characterizes the autooscillation process, which leads to a photometrical variability, the phase diagram corresponds to a limit cycle.For all stars studied, the limit cycles were computed. For a simple sine-like light curve, in e.g., L 2 Pup, the limit cycle is a simple ellipse. In a case of more complicated light curve, in which harmonics are statistically significant, the limit cycle has deviations from the ellipse.In an addition to a classical analysis, we use the error estimates of the smoothing function and its derivative to constrain an "error corridor" in the phase plane.
The studies of semiregular variables of stars by different authors are considered, and the main theoretical and observational problems associated with these stars are reviewed. Their evolutionary status and possible connection with long-period variables such as Mira Ceti are discussed.Individual objects belonging to different types of semiregular variables are described in detail.After leaving the main sequence, the stars pass through the region of instability of Cepheids, turning into radially pulsating variables of type δ Cepheus. These stars can be associated with semi-regular variables giants and supergiants of spectral classes F -K, which are usually denoted by the symbol SRd. In the process of further evolution of the variables of high luminosity fall in the region of red supergiants, becoming the type variables SRc (or Lc), and the variables lower luminosity turn into a semiregular variables SRab (or wrong Lb) of late spectral classes.Variables of the RV Tau type are a class of low-mass (with masses of the order of one solar) pulsating F -K − supergiants (M v = -3 m ÷ 5 m ), which may be at the short-term evolutionary stage of transition from the red giant to the protoplanetary nebula, which explains the small number of stars of this type of variability. Shklovsky (1956) was the first to point to stars of this type as the progenitors of planetary nebulae.
Periodogram analysis of 349 photovisual observations showed the peaks corresponding to periods 352 k 5d, 179.1 & 0.3d and 162.4 k 0.8d. The first value is very close to 1 year, so it might not be Justified by the present observations. The other two values obtained by 'two-parameter' search seem to be independent periods, but not the effect of 'observational windows'. No other significant peaks were detected in the interval from 65d to lood. Der erste Wert entspricht fast einem Jahr, so konnte er auf Grund der zeitlichen Beschrankung der vorliegenden Beobachtungen nicht verifiziert werden. Die beiden anderen Werte, durch eine ,,Zwei-Parameter"-Analyse gefunden, scheinen unabhangige Perioden zu sein und kein ,,Beobachtungsfenster"-Effekt. In dem Zeitraum von 65d bis 1 OOd konnten keine weiteren signifikanten Maxima nachgewiesen werden.Key worris: intrinsic variables -RX Bootis -maser AAA subject category: 122The star RX Bootis is a source of H,O maser emission (SCHWARTZ et al. 1974), and what is more surprising, no other maser emission (eg. OH) similarly to other objects was detected (BERULIS et al. 1983. VARDANJAN (1967VARDANJAN ( , 1978) discovered the polarisation of the optical emission, decreasing with the increasing wavelength (2 per cent at 1 = 0.45 pm and 0 per cent at 0.63 pm). He also detected the brightness change of 0.8" during 51d. However, the photometric behaviour of the star was studied very poorly, so the mean cycle length is believed to be in a wide interval from 78d (KEPING et al. 1984) to 3Md (very uncertain) in General Catalogue of Variable Stars (KHOLOPOV et al. 1985).The star was investigated on 349 photovisual plates taken in JD 2431 396-2445826 on the seven-camera astrograph of Astronomical Observatory of Odessa State University. The finding chart is shown at Fig. 1 where mi, i = 1 ... N , is the brightness of the star at the moment of time ti, w = 1/P is the frequency, P the trial period and mo, R and cp are the coefficients determined by the method of least squares. As the test function, the correlation coefficient r between the observed and calculated values according to eq. (1) was used. The corresponding periodogram is shown in Fig. 2. Three prominent peaks correspond to the periods 352,179 and 164days. The first value corresponds to the period published in GCVS (KHOLOPOV et al. 1985), but it is very close to 1 year, so the corresponding phase interval is covered by the present observations not completely. Thus the reality of the existence of this periodicity remains very questionable, and possibly it is an effect of "observational window". The part of the periodogram near the two other peaks is shown in Fig. 3.
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