  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 by Icke et al.
[Astron.Astrophys. 258, 341 (1992)] 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 Poincare 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 Poincare 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.
