  Recent remarkable progress in computing power and numerical analysis is
enabling us to fill a gap in the dynamical systems approach to turbulence. One
of the significant advances in this respect has been the numerical discovery of
simple invariant sets, such as nonlinear equilibria and periodic solutions, in
well-resolved Navier--Stokes flows. This review describes some fundamental and
practical aspects of dynamical systems theory for the investigation of
turbulence, focusing on recently found invariant solutions and their
significance for the dynamical and statistical characterization of
low-Reynolds-number turbulent flows. It is shown that the near-wall
regeneration cycle of coherent structures can be reproduced by such solutions.
The typical similarity laws of turbulence, i.e. the Prandtl wall law and the
Kolmogorov law for the viscous range, as well as the pattern and intensity of
turbulence-driven secondary flow in a square duct can also be represented by
these simple invariant solutions.
