The topics in the course vary from year to year and are typically chosen from: solitons, reaction-diffusion equations, numerical methods, bifurcation analysis, dimension reduction, Hamiltonian systems, fluid dynamics and perturbation theory.
This year, the course covers:
Modern computational techniques in non-linear dynamics and complex systems.
General course objectives:
Modern computational techniques such as continuation methods, numerical bifurcation analysis, dimension reduction, Fourier split step method and other methods for partial differential equations will be introduced. These allow to investigate those non-linear and complex systems for which analytical approaches are too difficult or even impossible to be used. Examples from physics, chemistry, biology and mechanical engineering will illustrate the different approaches and techniques. The exercises will cover analytical investigations of the approaches as well as implementing the computational methods in MATLAB.
Læringsmål:
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
characterize the long-term behaviour of dynamical systems
perform continuation of stationary solutions
perform continuation of periodic solutions
detect and classify bifurcation points
perform continuation of bifurcations points
use standard bifurcation software
augment standard bifurcation software
Kursusindhold:
The course covers modern numerical and analytical techniques to analyse and investigate non-linear differential equations and complex systems with many degrees of freedom.
Techniques will be presented to analyse low-dimensional non-linear systems. These are stability and growth investigations, continuation methods for steady states, numerical investigations for bifurcation points as well as their classification and identification by certain test functions. Furthermore, the continuation of periodic solutions, bifurcation points and points with optimality conditions will be treated. A particular focus of this course will be on using standard bifurcation software for analysing differential equations.
Further leading topics include:
Explicit reduction of high-dimensional complex systems to low-dimensional ones to enable analysis of these with the methods introduced before. Examples are the Karhunen-Loeve expansion (also known as principal component analysis or proper orthogonal decomposition), the centre manifold reduction and the slow manifold reduction.
Dimension reduction with Equation Free Methods to enable direct analysis of macroscopic (low-dimensional) properties of high-dimensional (microscopically defined) complex systems without requiring to perform an explicit reduction to a low-dimensional model. These so-called equation-free techniques or coarse analysis methods allow to investigate macroscopic model properties for which no closed mathematical equations are known, but are extremely important for scientific and engineering purposes. The behaviour on the macroscopic level (including numerical continuation and even bifurcation analysis) is analysed by switching between the macro and micro level while using short microscopic model evaluations in an advantageous way.
Litteratur:
References will be given during the lecture.
Bemærkninger:
The last three weeks of the semester will be dedicated to project work. Here you will work in small groups on problems that are relevant to engineering and science.
