Complex dynamics in a simple model of pulsations for super-asymptotic giant branch stars

Abstract

When intermediate mass stars reach their last stages of evolution they show pronounced oscillations. This phenomenon happens when these stars reach the so-called Asymptotic Giant Branch (AGB), which is a region of the Hertzsprung-Russell diagram located at about the same region of effective temperatures but at larger luminosities than those of regular giant stars. The period of these oscillations depends on the mass of the star. There is growing evidence that these oscillations are highly correlated with mass loss and that, as the mass loss increases, the pulsations become more chaotic. In this paper we study a simple oscillator which accounts for the observed properties of this kind of stars. This oscillator was first proposed and studied in [1] and we extend their study to the region of more massive and luminous stars -the region of Super-AGB stars. The oscillator consists of a periodic nonlinear perturbation of a linear Hamiltonian system. The formalism of dynamical systems theory has been used to explore the associated Poincaré map for the range of parameters typical of those stars. We have studied and characterized the dynamical behaviour of the oscillator as the parameters of the model are varied, leading us to explore a sequence of local and global bifurcations. Among these, a tripling bifurcation is remarkable, which allows us to show that the Poincaré map is a nontwist area preserving map. Meandering curves, hierarchical-islands traps and sticky orbits also show up. We discuss the implications of the stickiness phenomenon in the evolution and stability of the Super-AGB stars.

Creation of twistless circles in a model of stellar pulsations

“…In the absence of a better explanation, it was attributed to long-term (secular) nonlinear effects in the stellar envelopes. In a previous work [6], we have shown that our simple model recovers such a behaviour as it is generic of Hamiltonian systems. We also presented clear examples of sticky orbits together with an extensive discussion of their implications in the framework of the classification of variable stars.…”

“…In the present work we have extended the preliminary results of Ref. [6] concerning the dynamics of a forced oscillator as a model of irregular stellar pulsations. The driving is characterized by two parameters, the fractional amplitude of the internal perturbation α and the total amplitude of the driving ǫ.…”

A weakly non-adiabatic one-zone model of stellar pulsations: application to Mira stars

“…The values of ω used in their work were equivalent to adopting stellar models in the family of low‐mass stars ( M ≤ 5–8 M ⊙ ) reaching the AGB. In a recent publication (Munteanu et al 2002), we extended their conclusion to intermediate‐mass stars (8 M ⊙ ≤ M ≤ 11 M ⊙ ) also in the AGB phase, more precisely to values of ω around 3. In the previous sections, we have shown that in the case of ω= 20.1 a peculiar behaviour is born from the interplay between non‐adiabaticity and internal perturbation.…”

“…The one‐zone model approach is especially suited for AGB stars as the density difference between the central core and the outer layers is so large that these two regions can be considered as effectively decoupled. Following Icke, Frank & Heske (1992) and Munteanu et al (2002), we consider the stellar pulsation to be simulated by a variable inner boundary located well beneath the photosphere and moving with constant frequency (piston approximation). This sinusoidal driving consists of pressure waves originating in the interior and propagating through a transition zone until they dissipate in the outer layers.…”

Meanders and reconnection–collision sequences in the standard nontwist map

“…[2] Some basic concepts used throughout the paper are reviewed in Appendix A. Nontwist maps are used to describe many physical systems, e.g., magnetic field lines in tokamaks (see, e.g., Refs. [3][4][5][6][7][8][9]) and stellarators [10,11] (plasma physics); planetary orbits, [12] stellar pulsations [13] (astronomy); traveling waves, [1,14] coherent structures and selfconsistent transport [15] (fluid dynamics). Additional references can be found in Refs.…”