Nonlinear Dynamics and Complex Systems 2

Module PH2028

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Basic Information

PH2028 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

  • General catalogue of special courses
  • Specific catalogue of special courses for Applied and Engineering Physics
  • Specific catalogue of special courses for Biophysics
  • Specific catalogue of special courses for condensed matter physics

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workloadContact hoursCredits (ECTS)
150 h 75 h 5 CP

Responsible coordinator of the module PH2028 is Katharina Krischer.

Content, Learning Outcome and Preconditions


This module provides an introduction to self-organization and pattern formation in spatially extended systems. After a motivation in which the universality of the observed patterns and their unified mathematical description are elucidated, the basic mechanisms that lead to spatio-temporal self-organization are discussed. We mainly focus on reaction-diffusion systems. The phenomena considered are ordered according to their complexity. First traveling waves in one-component bistable systems are explored, then pulses and spiral waves in excitable systems are discussed. Subsequently, we study the formation of Turing structures in spatially one and two-dimensional systems. Finally, oscillatory dynamics is considered. Here we begin by looking at an ensemble of globally coupled oscillators, elucidating the so-called Kuramoto transition from incoherent behavior to synchronized oscillations in detail, and then discuss synchronization behavior of oscillatory networks in a general context. Thereafter, the complex Ginzburg-Landau equation as prototypical equation for diffusively coupled oscillatory media is introduced, and the transition to spatio-temporal chaos investigated.

Learning Outcome

After participation in the Module the student is able to

  1. understand the basic mechanisms that lead to patterns and cooperative phenomena in dissipative systems far from the thermodynamic equilibrium
  2. explain the universal laws leading to pattern formation in reaction-diffusion systems in the bistable excitable and oscillatory regime with prototypical models
  3. explain the origin of synchronization phenomena in coupled oscillatory networks
  4. perform simulations of reaction-diffusion system and classify the observed patterns.


Nonlinear Dynamics and Complex Systems I (recommended but not essential)

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

VU 4 Nichtlineare Dynamik und komplexe Systeme 2 Krischer, K.
Mitwirkende: Heger, D.
Donnerstag, 10:00–12:00
sowie Termine in Gruppen

Learning and Teaching Methods

lecture, beamer presentation, board work, exercises in individual and group work


practise sheets, accompanying internet site, complementary literature


  • Lecture Script
  • A.S. Mikhailov, "Foundations of Synergetics I"
  • G. Nicolis, "Introduction of Nonlinear Science"
  • J. D. Murray "Mathematical Biology II"

Module Exam

Description of exams and course work

In an oral exam the learning outcome is tested using comprehension questions and sample problems.

In accordance with §12 (8) APSO the exam can be done as a written test. In this case the time duration is 60 minutes.

Exam Repetition

There is a possibility to take the exam at the end of the semester. There is a possibility to take the exam in the following semester.

Condensed Matter

When atoms interact things can get interesting. Fundamental research on the underlying properties of materials and nanostructures and exploration of the potential they provide for applications.

Nuclei, Particles, Astrophysics

A journey of discovery to understanding our world at the subatomic scale, from the nuclei inside atoms down to the most elementary building blocks of matter. Are you ready for the adventure?


Biological systems, from proteins to living cells and organisms, obey physical principles. Our research groups in biophysics shape one of Germany's largest scientific clusters in this area.