Discrete Map Simulation of Stellar Oscillations

Abstract

In order to isolate the classes of nonlinear mono-mode stellar oscillations, we introduce a simulation of the pulsation dynamics by a parametrized iterative algebraic map whose structure is derived from the equations of stellar fluid dynamics. We report here on an extensive numerical investigation of the role of 3 of the parameters of the map. Among the latter the linear growth-rate times the linear period is found to play a dominant part: Under well specified conditions this parameter generates a Feigenbaum period-doubling sequence. We further show that the 3-dimensional free parameter space supports period-K oscillations, K any integer. While P-1, P-2, P-4 and P-8 oscillations occupy most of the parameter space over which the oscillations are regular, P-3 and P-6 as well as P-5 and P-7 oscillations are carried by smaller zones. Our numerical results are also indicative that the regular region in the parameter space is crisscrossed by a web of chaotic behaviour. A numerical investigation of the geometry of the region in the 3-dimensional free parameter space sustaining regular oscillations indicates that the boundary of this zone has a locally irregular fractal-like structure which alternates with an apparently smooth structure. We argue that this result implies that the zone of regular oscillations in the HR-diagram shows a fractal-like boundary in some regions while in other parts of the HR-diagram this boundary is smooth. Accordingly an evolutionary track of a star will cut the boundary of regular behaviour a large number of times if the crossing occurs near a locally fractal-like stretch. Our iterative model is thus consistent with the observation of a variety of transitions from regular oscillations to chaos and from chaotic oscillations back to regular behaviour in variable stars.