Nonlinear dynamical systems, with emphasis on discrete time systems (mappings), in one, two and three dimensions. The logistic map, models to predict population dynamics. Mechanical systems that show unpredictable behaviour (e.g. the double pendulum), stability properties and periodic orbits, graphic analysis. Routes from order to chaos via: (1) period doubling bifurcations, (2) intermittency and (3) the break-up of quasi-periodic orbits. Analysis of the relevant bifurcation types: pitchfork, transcritical, saddle-node and (in the context of limit cycles) the Hopf bifurcation. The method of renormalization and the "universal" constants of Feigenbaum. Strange (chaotic) attractors and mappings in more than one dimension. The models of Hénon and Lorenz. Fractal sets, box-counting dimension and Hausdorff dimension. The triadic fractal set of Cantor. The fractal triangular set of Sierpinski and its relation to the so-called Chaos Game. Koch's snowflake. Fractals with multiple scaling factors (multifractals) and the theory of generalized dimensions. Julia sets, the Mandelbrot set. Invariant sets, symbolic dynamics and the theory of chaos according to Smale. Nonlinear analysis of chaotic time series, with applications in the natural sciences.