The search for extrasolar planets is one of the most exciting fields in astronomy and will perhaps one day answer the question of whether we are alone in the universe. The exploration of extrasolar planets makes it possible to put the formation and evolution of our Solar System into a more generalized cosmogonic context.

Long-period Variables and Miras are radially pulsating, highly evolved stars on the Asymptotic Giant Branch. Due to peculiar conditions of these objects, their cool, extended atmospheres are distinguished sites for the formation of complex molecules and dust particles. The interplay between dust formation and stellar radiation results in a circumstellar dust shell (CDS) generating slow mass loss, which finally enriches the interstellar medium with processed material.
In view of the theory of dynamical systems, the CDS can be considered as nonlinear multi-oscillatory systems, whose eigenfrequencies and normal modes are controlled by the intrinsic time scales of various coupled physical and chemical processes. I investigate the complex dynamical behaviour of carbon-rich CDSs in great detail by established methods of non-linear dynamics such as Fourier Analysis and the study of stroboscopic and Poincaré maps.