Two different models of open, dissipative systems are considered. The stability and the asymptotic manifold of solutions are studied with respect to the variation of a model viscosity ν. Bifurcation theorems result from the analysis which shows the existence of a class of ‘prototurbulent’ motions. These are generalisations of the quasi-periodic asymptotic motions characteristic of Hopf's well-known mathematical model of turbulence. Given the statistics of the driving external field, it is shown how to calculate the correlation functions of the asymptotic motions (as t → ∞) for arbitrary values of the model viscosity. The systems treated are non-linear and except in special cases, cannot be solved in closed form. The relation to real systems is discussed. The effects of dimensionality on the qualitative nature of the motion is explored. It is found that for the models treated, perturbations of the external field do not affect the qualitative properties. Certain ‘mean’ quantities vary smoothly with the variation of the external field, but the details of the motion may be sensitive to the external perturbations. It is also found that even if the motion is strictly not quasiperiodic in the sense of Hopf and Landau, almost all asymptotic motions have a recurrent character under certain conditions. Some of the models also show the reverse of this behaviour, i.e. a non-dissipative, quasi-recurrent approach to ‘equilibrium’. This phenomenon can occur if the number of active degrees of freedom at a non-zero model viscosity is infinite.