By Pierre Collet
The research of dynamical platforms is a good tested box. This booklet presents a landscape of numerous elements of curiosity to mathematicians and physicists. It collects the cloth of a number of classes on the graduate point given via the authors, fending off specific proofs in trade for various illustrations and examples. except universal topics during this box, loads of recognition is given to questions of actual size and stochastic houses of chaotic dynamical platforms.
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Additional resources for Concepts and Results in Chaotic Dynamics: A Short Course (Theoretical and Mathematical Physics)
Local stable and unstable manifolds around a 2-dimensional hyperbolic fixed point, x0 Proof (sketch). 9, and refer to (Hirsch, Pugh, and Shub 1977) for a complete proof and extensions. The basic idea of the proof goes as follows: The local stable manifold is constructed as a piece of graph of a map from the stable space to the unstable space. More precisely, using near the fixed point (assumed to be at the origin) coordinates given by the decomposition E s ⊕E u , one looks for a map g from E s to E u whose graph is invariant, namely f η, g(η) = η , g(η ) for any η in a neighborhood of 0 in E s , and η some point in E s (depending on η).
Some popular examples are maps of the interval 0, 231 − 1 given by f (x) = 16,807 x mod 231 − 1 or f (x) = 48,271 x mod 231 − 1 which are among the best generators for 32 bits machines. We refer to (Knuth 1981; L’Ecuyer 1994; 2004), and (Marsaglia 1992) for more on the subject and for quality tests. 18. The Logistic Map The previous dynamical systems have discontinuities, and many examples have been studied which are more regular, but often more difficult to analyze. One of the most well-known examples is provided by the one-parameter family of quadratic maps.
In particular, it attracts transversally all the orbits. For some partial differential equations, the existence of such inertial manifolds has been proven. Unfortunately, this is still an open question for the 2-dimensional Navier–Stokes equation. Another drawback is that often one is unable to show that these inertial manifolds are regular enough. The picture is however quite nice: there is a finite dimensional manifold that attracts all the trajectories (at least locally) and which therefore contains the asymptotic dynamics.