Point (ECTS )

The topics in the course are varying each year, and are typically chosen from:
solitons, reaction-diffusion equations, numerical methods, bifurcation
analysis, dimension reduction, Hamiltonian systems, fluid dynamics.

This year, the course covers:

Modern computational techniques in nonlinear 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 nonlinear 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.

• characterize the dynamical behaviour of dynamical systems
• perform continuation of stationary solutions
• perform continuation of periodic solutions
• detect and classify bifurcation points
• perform continuation of bifurcations points
• perform a dimension reduction by e.g. Karhunen-Loeve expansion
• reaction diffusion equations for bio systems
• Fourier split step method
• write own computer programs to perform the above mentioned approaches
• investigate analytically certain properties of the system
• multiscale perturbation theory

The course covers modern numerical and analytical techniques to analyze and
investigate nonlinear differential equations and complex systems with
many degrees of freedom.

First, techniques will be presented to analyze low-dimensional
nonlinear 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 testfunctions. Furthermore, the continuation of periodic
solutions, bifurcation points and points with optimality conditions
will be treated.

Second, high-dimensional complex systems will be reduced to
low-dimensional ones which allows then in the next step to analyze
these with the methods introduced before. Examples are the
Karhunen-Loeve expansion (also known as principal component analysis
or proper orthogonal decomposition) and the slow manifold reduction.

Third, it will be covered how to analyze directly the macroscopic
(e.g. low-dimensional) properties of high-dimensional
(i.e. microscopically defined) complex systems without requiring to
perform first a reduction to a low-dimensional description. 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 in
an advantageous way short microscopic model evaluations. The space of
variables and parameters is just too large and the computational costs
are therefore even for a parallel cluster too expensive to be able to
obtain similar information by ``brute force'' direct simulations.

References will be given during the lecture.