Ruelle-Takens-Newhouse Route

This route, due to Ruelle and Takens [ruelle71], assumes that the onset to turbulence starts out exactly as in the Landau-Hopf scenario, namely by an increase in frequency components. Unlike that scenario, however, Ruelle and Takens suggested 

How does this relate to fluid turbulence The idea is that there exists a critical value of the Reynolds number, TZe, such that intermittent turbulent behavior can appear in the system for TZ TZe- Moreover, if the behavior of the Lorenz system correctly identifies the underlying mechanism, it may be predicted that, as TZ changes, (1) the duration of the intermittently turbulent behavior will be random, and (2) the mean duration of the laminar phases in between will vary as 

Lorentz found that when r = 28, cr = 10 and b = 8/.1, (i) the trajectories are attracted to a bounded region in space, (ii) the motion appears chaotic, and (iii) there is a great sensitivity to initial conditions. 

Both type-I intermittency [berge80] and type-II intermittency [duboisS] has been observed in a Rayleigh-Benard experiment, Generally speaking, however, only a relatively small subset of experiments appear to be consistent with this particular route, 

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Figure 12.10. Ruelle-Takens newhouse route to chaos (a) periodic (b) biperiodic (c) period-4 and (d) chaos.

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