Landau proposed to explain the onset of turbulence in terms of successive apparition of limit cycles. Combined limit cycles will in general produce quasi-periodic motion. We have seen that this type motion has no sensitive dependence on the initial conditions. This feature was criticized by Ruelle and Takens who introduced the notion Strange Attractors for which this sensitivity is essential. We will not attempt to define abstractly the notion of strange attractor, but rather give two example.
The Lorenz Model:
Figure 2: The Lorenz attractor for 
, b=8/3 and r=166.2.
The Henon Model
This is a model with discrete time (as what we could obtain for a Poincaré section). The parameter b controls the ``shrinking', and the parameter a the ``stretching'' (see Henon, Communications in Mathematical Physics 50 p. 69 (1976)). For a=1.4 and b=0.3, the system is trapped along a ``strange'' figure. Unlike for quasiperiodic motion, there is a sensitive dependence in the initial conditions.
Figure 3: The Henon attractor a=1.4 and b=0.3.
Experimental Results
The only possible way to decide which scenario for the onset of turbulence is correct is to make experiments. In the case of layer of fluid heated from below, it is possible to measure for instance the vertical component of the velocity at a given point and at successive instants. These time series can be analyzed using the Fourier Transform.