  The recently proposed definition of complexity for static and spherically
symmetric self--gravitating systems [1], is extended to the fully dynamic
situation. In this latter case we have to consider not only the complexity
factor of the structure of the fluid distribution, but also the condition of
minimal complexity of the pattern of evolution. As we shall see these two
issues are deeply intertwined. For the complexity factor of the structure we
choose the same as for the static case, whereas for the simplest pattern of
evolution we assume the homologous condition. The dissipative and
non-dissipative cases are considered separately. In the latter case the fluid
distribution, satisfying the vanishing complexity factor condition and evolving
homologously, corresponds to a homogeneous (in the energy density), geodesic
and shear-free, isotropic (in the pressure) fluid. In the dissipative case the
fluid is still geodesic, but shearing, and there exists (in principle) a large
class of solutions. Finally we discuss about the stability of the vanishing
complexity condition.
