Nonlinearity and nonlinear world. The sources of nonlinearities due to physics and geometry. Nonlinear mathematical models. Basic theory of ODEs. Attractors, bifurcations. Mathematically determined chaos. Feigenbaum diagram, Lorenz section, Poincaré section. Fractality, fractal structures. Recurrence maps. Mandelbrot set and Julia sets. Multibrot sets and nonlinear dynamical systems. Fractal dimensions. Universal route to chaos. Identification of chaotic processes. Analytical and numerical methods, Lyapunov exponent. Entropy. Horizon of predictability.
Basic principles of modelling of natural and engineering processes. Predictability, methods of modelling, basic types (statical, dynamical). Mathematical models. Dimensional analysis and similarity. Design of experiments. Properties of non-linear problems, the loss of predictability. Practical examples. Contemporary ideas on modelling (Prigogine, Nicolis) complexity and simplicity.
Examples from physics, mechanics, biology and ecology. Applications of chaos theory and fractal geometry.
- Õpetaja/Teacher: Dmitri Kartofelev
- Õpetaja/Teacher: Tanel Peets