  We study the changes in the dynamical behaviour of axisymmetric spherical
mean-field dynamo models produced by changes in their geometry and topology, by
considering a two parameter family of models, ranging from a full sphere to
spherical shell, torus and disc-like configurations, within a unified
framework. We find that the two parameter space of the family of models
considered here separates into at least three different regions with distinct
characteristics for the onset of dynamo action. In two of these regions, the
most easily excited fields are oscillatory, in one case with dipolar symmetry,
and in the other with quadrupolar, whereas in the third region the most easily
excited field is steady and quadrupolar. In the nonlinear regime, we find that
topological changes can alter significantly the dynamical behaviour, whilst
modest changes in geometry can produce qualitative changes, particularly for
thin disc-like configurations. This is of potential importance, since the exact
shapes of astrophysical bodies, especially accretion discs and galaxies, are
usually not precisely known.
