A Study of Period Doubling in a One-Zone Pulsating Stellar Model

Abstract

A simple one-zone model for nonlinear stellar pulsations is outlined and applied to the study of period doubling observed in some W Virginis and RV Tauri stars. The model reveals a number of period doubling bifurcations as the parameters are varied, similar to those found by Buchler & Kovacs in their series of hydrodynamic models. In the vicinity of a stable limit cycle, despite its large number of degrees of freedom, the model develops an essentially one-dimensional Poincaré’s return map, determining the modulation of the amplitude. The analysis of these maps confirmed that period doubling has its origin in a strong nonlinear increase of total energy losses per period as radial amplitude, δr/r, increases. An additional study of nearly periodic hydrodynamic models with P > 15 days, calculated including radiative transfer effects, shows that the rate of energy dissipation per period by shocks in the atmospheres increases rapidly with δr/r, whereas the excitation rate, δ_o, remains rather stable. This permitted us to construct an analytic return map for maxima of the total kinetic energy which clearly demonstrates the mechanism of successive period doubling as δ_o, the sole bifurcation parameter, grows monotonically.