# Nonlinear Dynamics and Pattern Formation

summer term 2021
Lecturer: Ziebert
53 participants

Note: Due to the current corona situation, both the lecture and the tutorials will be held as online meetings.

The lectures are on Monday and Wednesday 2:00pm on zoom (see link above).

Motivation:

Nonlinear dynamics is an interdisciplinary part of mathematical physics, with applications in such diverse fields as mechanics, optics, chemistry, biology, ecology, to name but a few. Equations with nonlinearities show a much more diverse behavior than their linear counterparts, for instance self-sustained oscillations, nonlinear competition (as linear superposition does not hold anymore), chaotic dynamics and pattern formation. Pattern formation, in turn, is one of the most fascinating and intriguing phenomena in nature: it takes place in a wide variety of physical, chemical and biological systems and on very different spatial and temporal scales: examples are convection phenomena in geosciences and meteorology, but also patterns occurring in chemical reactions and bacterial colonies. In some circumstances, pattern formation is undesired, for instance the formation of spiral waves leading to cardiac arrhythmias in the heart muscle. In other contexts, pattern formation is even essential for the functioning of a system as in cell division and embryo development.

Contents:

The lecture will start with an introduction to nonlinear dynamics on the level of ordinary differential equations (ODEs), introducing concepts like phase space analysis, attractors, (in)stability of solutions and bifurctions, as well as nonlinear oscillations.

We will then proceed to study spatio-temporal behavior, i.e. partial differential equations (PDEs) and discuss the main questions in pattern formation: when will a homogeneous state become structured, i.e. unstable towards a pattern? What are the generic scenarios/types of patterns? When are patterns stable and are they unique? What determines the wavelength / period in time / amplitude of a pattern? Importantly, a universal description of pattern dynamics exists, that is independent of the system-specific pattern formation mechanism. The method to obtain this description is called multiple-scale reduction, resulting in an amplitude equation (also called center manifold), which is nothing but the famous Ginzburg-Landau equation (Nobel Prize in Physics 2003, originally derived for superconductivity).

Finally, nonlinear waves and solitons (localized waves) will be discussed. They again occur in many systems, from coupled nonlinear springs to hydrodynamic surface waves and nonlinear optics. In addition, solitons have intrigueing mathematical properties that will also be discussed.

Prerequisites:

The course is designed for physics students in advanced bachelor and beginning master semesters (students from other disciplines are also welcome). It will be given in English. A basic understanding of physics and differential equations is sufficient to attend. Exercises will be discussed in a tutorial (time and place to be defined, please register).

Literature:
- SH Strogatz, Nonlinear dynamics and chaos, Westview 1994
- Cross M C and Hohenberg P C, Rev. Mod. Phys. 1993.
- Cross M C and Greenside H, Pattern formation and dynamics in nonequilibrium systems (Cambridge,
Cambridge Univ. Press, 2009).

## Exercise sheets

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## Practice groups

Nonlinear Dynamics and Pattern Formation
summer term 2021
Ziebert