Courses

Nonlinear dynamical systems, with emphasis on discrete-time systems (imagings), one, two and three dimensions. The logistic representation and the population evolution prediction model. Mechanical examples of unpredictable behavior (e.g. the double pendulum), stability and periodic trajectories, graphical analysis. Transition to chaos through: (1) Period doubling bifurcations, (2) Intermittency and (3) Breakdown of quasi-periodic trajectories. Analysis of the corresponding bifurcations: bifurcation, postcritical, sagittal-node and limit cycle bifurcation (Hopf). Renormalization method and Feigenbaum's "global" constants. Strange (chaotic) attractors and representations in two or more dimensions. The models of Hénon and Lorenz. Fractals, their capacity dimension and the Hausdorff dimension. Cantor's ternary set. Sierpinski's triangle and the so-called Chaos Game. Koch's flake and square. Multifractals and the theory of generalized dimensions. Julia sets, Mandelbrot set. Invariant sets, symbolic dynamics and Smale's chaos theory. Nonlinear analysis of chaotic time series and applications in the sciences.