Period Doubling with Hysteresis in BL Her-Type Models

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 only T_eff 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 to 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 T_eff (typically 100-150K), in which regular solution becomes unstable towards a priod 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. This is the same resonance, which also causes period doubling in the models of classical Cepheids. 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. We show the bifurcation for one of our sequences, by plotting minimum pulsational velocities V_min versus T_eff.