WEAK CHAOS IN LONG-PERIOD VARIABLES

When low- to intermediate mass stars reach the very late stages of their evolution, they show long period oscillations. There is an increasing body of observational evidence which indicates that these oscillations become more and more irregular as a star begins to leave the asymptotic giant branch (AGB); that mass loss accompanies these oscillations, and that the mass loss rate increases progressively as the oscillations become more erratic; and that the mass loss can be very aspherical. We have attempted to account for the increasing irregularity of the long-period variations by investigating how the outer layers ("mantle") of an evolved AGB star respond to pulsations which originate in the stellar interior. We assume that the oscillations arise in a zone of instability well below the stellar photosphere. The mantle of the star is driven periodically by pressure waves generated by the interior motion. This type of driven oscillator exhibits chaotic motions for a wide range of relevant parameters. A rather remarkable thing about this chaos mechanism is, that chaotic motions occur even though the unperturbed equation of motion of the stellar mantle contains no hyperbolic singularities or rational resonances. This implies that a perturbation cannot produce chaos via the familiar route indicated by the Poincare-Birkhoff and Kolmogorov-Arnol'd-Moser theorems. Instead, we obtain "weak" chaos, in which the hyperbolic points which are a prerequisite for chaos are caused by the perturbation itself. We study our driven-oscillator equations in detail, in various approximations. Analytic estimates show why and where chaos ought to set in; extensive numerical experiments confirm and extend these findings. Because the driven-oscillator equation is quite simple (it embodies only the bare essentials of the dynamics of oscillating stellar mantles), we present only an excerpt of our numerical findings, while encouraging the reader to verify these rather straightforward numerical experiments. When comparing the outcome of our analysis with observations, we note that the characteristics mentioned above can be traced directly to the dynamics of a driven oscillator: motions are regularly periodic at first, then become multiperiodic, and finally chaotic, while the mass of the mantle decreases more and more rapidly through mass loss.