Mathematical aspects of the theory of development of turbulence in the sense of Landau

This paper contains some rigorous mathematical results related to the theory of development of turbulence in the sense of Landau. In diverse areas of the natural sciences, specific examples of non-linear dynamical systems are considered (including E. Hopf's classical example) whose attractors turn out to be invariant tori of arbitrarily high dimension under an appropriate change of parameters. The investigation of these examples enables us to give a rigorous meaning to the notion of a ‘turbulent attractor’ in some cases and to reveal the main properties of such an attractor, notable among which are its fractal property and its infinite dimensionality.