Abstract
I want to describe some joint work with Gérard IOOSS of Nice University connected with the Ruelle and Takens deterministic approach to turbulence. The main references are [R.T.], [C.I. 1.], [C.I.2. ].
Navier-Stokes equations can be thought of as a flow in some infinite dimensio-nal Banach space of divergence-free vector-fields in ℝ3 with some Sobolev norm (for regularity properties analogous to those found in the finite dimensional case, even though we deal with unbounded operators, see [I]. As a certain parameter varies (for ex. the Taylor number in Couette-Taylor experiment] the flow is at first supposed to undergo a sequence of bifurcations of the Hopf type, eventually leading to a stable (this means here asymptotically stable) invariant r-torus, r not too big say r = 4).
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Bibliography
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Chenciner, A. (2010). Hopf Bifurcation for Invariant Tori. In: Marchioro, C. (eds) Dynamical Systems. C.I.M.E. Summer Schools, vol 78. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13929-1_4
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