By Étienne Ghys (auth.), Bertrand Duplantier, Stéphane Nonnenmacher, Vincent Rivasseau (eds.)

This 12th quantity within the Poincaré Seminar sequence offers an entire and interdisciplinary standpoint at the thought of Chaos, either in classical mechanics in its deterministic model, and in quantum mechanics. This e-book expounds the most vast ranging questions in technology, from uncovering the fingerprints of classical chaotic dynamics in quantum platforms, to predicting the destiny of our personal planetary method. Its seven articles also are hugely pedagogical, as befits their beginning in lectures to a huge clinical viewers. Highlights comprise a whole description through the mathematician É. Ghys of the paradigmatic Lorenz attractor, and of the famed Lorenz butterfly impact because it is known this day, illuminating the basic mathematical concerns at play with deterministic chaos; a close account by means of the experimentalist S. Fauve of the masterpiece scan, the von Kármán Sodium or VKS scan, which confirmed in 2007 the spontaneous iteration of a magnetic box in a strongly turbulent stream, together with its reversal, a version of Earth’s magnetic box; an easy toy version through the theorist U. Smilansky – the discrete Laplacian on finite *d-*regular expander graphs – which permits one to know the basic elements of quantum chaos, together with its primary hyperlink to random matrix idea; a evaluate by means of the mathematical physicists P. Bourgade and J.P. Keating, which illuminates the attention-grabbing connection among the distribution of zeros of the Riemann ζ-function and the statistics of eigenvalues of random unitary matrices, which may eventually offer a spectral interpretation for the zeros of the ζ-function, hence an explanation of the distinguished Riemann speculation itself; a piece of writing by means of a pioneer of experimental quantum chaos, H-J. Stöckmann, who exhibits intimately how experiments at the propagation of microwaves in second or 3D chaotic cavities superbly make certain theoretical predictions; an intensive presentation through the mathematical physicist S. Nonnenmacher of the “anatomy” of the eigenmodes of quantized chaotic structures, particularly in their macroscopic localization houses, as governed by way of the Quantum Ergodic theorem, and of the deep mathematical problem posed through their fluctuations on the microscopic scale; a evaluation, either old and clinical, by way of the astronomer J. Laskar at the balance, for this reason the destiny, of the chaotic sun planetary method we are living in, a subject matter the place he made groundbreaking contributions, together with the probabilistic estimate of attainable planetary collisions. This e-book might be of extensive basic curiosity to either physicists and mathematicians.

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He conjectures that property to be generic. Smale’s 1967 article Diﬀerential dynamical systems represents an important step for the theory of dynamical systems [76], a “masterpiece of mathematical literature” according to Ruelle [69]. But, already in 1968, Abraham and Smale found a counterexample to this new conjecture, also showing that Axiom A is not generic [1]. In 1972, Shub and Smale experiment another concept of stability [70], which will lead Meyer to the following comment: In the never-ending quest for a solution of the yin-yang problem more and more general concepts of stability are proﬀered.

The central part illustrates a step in the deformation. For more explanations, see [26, 28]. Figure 22. Deformation of a modular knot into a Lorenz knot 40 E. Ghys The analogy between a ﬂuid motion and a geodesic ﬂow is not new. In a remarkable 1966 article, Arnold showed that the Euler equation for perfect ﬂuids is nothing but the equation for the geodesics on the inﬁnite-dimensional group of volume preserving diﬀeomorphisms [7]. Besides, he shows that the sectional curvatures are “often” negative, inducing him to propose a behavior ` a la Hadamard for the solutions of the Euler equation.

A Lorenz geometric model come back inside the square ????. One then obtains a vector ﬁeld deﬁned only inside a certain domain ???? sketched in Figure 15. The main objective is to understand the dynamics inside ????, but one can also extend the vector ﬁeld outside this domain to get a globally deﬁned ﬁeld. Such a vector ﬁeld ???? is a geometric Lorenz model. f Figure 16. Return maps An important remark is that not all the points of the triangles originate from the square: the tips do not, since they come from the singular point at the origin.