Chaotic dynamical systems (2/6)
Stéphane Nonnenmacher
Fri, Sep. 25th 2009, 10:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
The aim of this course is to present some properties of low-dimensional dynamical systems, particularly in the case where the dynamics is ``chaotic''. We will describe several aspects of ``chaos'', by introducing various ``modern'' mathematical tools, allowing us to analyze the long-time properties of such systems. Several simple examples, leading to explicit computations, will be treated in detail. A tentative plan (not necessarily chronological) follows. \\ \par \noindent - Definition of a dynamical system: flow generated by a vector field, discrete time transformation. Poincaré sections vs. suspended flows. Examples: Hamiltonian flow, geodesic flow, transformations on the interval or on the two-dimensional torus. \\ - Ergodic theory: long-time behavior. Statistics of long periodic orbits. Probability distributions invariant through the dynamics (invariant measures). ``Physical'' invariant measure. \\ - Chaotic dynamics: instability (Lyapunov exponents) and recurrence. From the hyperbolic fixed point to Smale's horseshoe. \\ - Various levels of chaos: ergodicity, weak and strong mixing. - Symbolic dynamics: subshifts on 1D spin chains. Relation (semiconjugacy) with expanding maps on the interval. \\ - Uniformly hyperbolic systems: stable/unstable manifolds. Markov partitions: relation with symbolic dynamics. Anosov systems. Example: Arnold's ``cat map'' on the two-dimensional torus. \\ - Complexity theory. Topological entropy, link with statistcs of periodic orbits. Partitions functions (dynamical zeta functions). Kolmogorov-Sinai entropy of an invariant measure. \\ - Exponential mixing of expanding maps: spectral analysis of some transfer operator. Perron-Frobenius theorem. \\ - Structural stability vs. bifurcations. Examples: logistic map on the interval/non-linear perturbation of the ``cat map''. \\ \\ \\ (Cour organisé en collaboration avec l'Ecole Doctorale de Physique de la Région Parisienne - ED 107)
Contact : Riccardo GUIDA


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