  Long-period variables (LPVs) and Miras are highly evolved stars at the upper
region of the asymptotic giant branch. They exhibit a pronounced variability of
their luminosity with a more or less well defined period and suffer large mass
loss in the form of stellar winds. Due to this extensive mass loss, LPVs and
Miras are surrounded by extended circumstellar dust shells (CDSs). The dynamics
of these envelopes is the result of a complex interplay via an external
excitation by the pulsating central star, dust formation and radiative
transfer. Our study is aimed at an understanding of the dynamics of CDSs around
carbon-rich, standard luminous LPVs and Miras. These shells often show
multiperiodicity with secondary periods as high as a few 10^4 d superimposed on
a main period which is in the range of approximately 10^2-10^3 d. Such
secondary periods may be caused at least in part by the presence of dust. We
consider an excitation of the CDSs by either a harmonic force, provided by the
oscillation of the central star, or by a stochastic force with a continuous
power spectrum. The resulting numerically computed dynamical behaviour of the
shell is analysed with the help of Fourier analysis and stroboscopic maps. CDSs
may be described as multioscillatory systems which are driven by the pulsating
stars. A set of normal modes can be identified. The obtained periods of these
modes are of the order of some 10^3 d, which is a characteristic timescale for
dust nucleation, growth and elemental enrichment in the dust formation zone.
Depending on the oscillation period and strength of the central star, the
envelope reacts periodically, multi- periodically or irregularly.
