MA 191 k Topics in Mathematical Chaos

 This is an introduction into the Mathematics of chaos. This course was given in in the second term 1994 at Caltech. The Dvi file (223 pages, no pictures but with exercices) of the course is is still available. It has only once been revised since the material was handed out to the participants of the course. The material was quite dense. Some students mentioned at the end that it was too dense for one term (10 weeks)). Proofs were given in class. To the content: Week 1: Dynamical systems in general (A bluffer's guide to chaos, The too abstract point of view, The basic questions, On the definition of "chaos", Examples of dynamical systems. Week 2: Henon maps (The shift, Contractions in metric spaces, The implicit function theorem, Embedding a shift in the Henon map, Existence of the stable and unstable manifolds, a strange attractor?, Epilog and open research problems. Week 3: Fractals (Hausdorff dimension and fractals, Properties of the Hausdorff dimension, The Hutchinson operator, Examples, The box counting dimension, The similarity dimension, Dynamical systems on fractals, Skewproduct of contractions, The Hutchinson operator on measures, Overview over different dimensions Week 4: Billiards (The Billiard map, the generating function, existence of periodic orbits, classification of periodic orbits, formula of Green-Mackay-Meiss, Caustics, Lyapunov exponents, Positive Lyapunov exponents for the stadium, Exterior billiards, Open research problems) Week 5: Rational maps (Embedding a one-sided shift in the quadratic map, Special parameter values, The Mandelbrot set, The Julia sets, Preliminaries in complex analysis, The connectivity of the Mandelbrot set, Local dynamics near fixed points, The Green function, M is connected, The Mandelbar set and other sets, Iteration of rational maps, Embedding of a one-sided shift in rational maps, research problems. Week 6: Standard maps (The standard map, Generalized standard maps, The generating function, Periodic orbits, Embedding a shift, The KAM theorem (without proof), Aubry-Mather sets, An analytic map, Topological entropy, Hyperbolic sets, Measuring the Lyapunov exponents, Search for little elliptic ilands. Week 7: Cellular Automata (Definition of cellular automata, Topological entropy of cellular automata, Examples of cellular automata, A map on subshifts, Properties of subshifts invariant under CA maps, Minimal subshifts, Turing machines as dynamical systems, Dynamical systems with higher dimensional time, Open research problems) Week 8: Coupled map lattices (Embedding a two dimensional shift in coupled map lattices, Discrete neural networks, Overview, Open research problems) Week 9: Differential equations (The Lorenz system, An existence theorem for differential equations, Global existence of the Lorenz flow, The Poincare Bendixon theorem, Equilibrium points, Periodic orbits and knots, Poincare sections, The "strange" attractor, Reconstruction of attractors from datas, measurements, Some limiting cases, Other systems) Week 10: N-body problems (The celestial N body problem, Vortex systems, Toda systems) Week 11: Dynamical systems in number theory (The Collatz problem, Encryption algorithms, The discrete logarithm problem, recurrence and the theorem of van der Waerden)