Nonlinear Dynamics and Complex Systems 2
Module version of SS 2018
There are historic module descriptions of this module. A module description is valid until replaced by a newer one.
Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.
|available module versions|
|SS 2020||SS 2019||SS 2018||SS 2017||SS 2011|
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.
- Specific catalogue of special courses for condensed matter physics
- Specific catalogue of special courses for Biophysics
- Specific catalogue of special courses for Applied and Engineering Physics
- Complementary catalogue of special courses for nuclear, particle, and astrophysics
- Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)
If not stated otherwise for export to a non-physics program the student workload is given in the following table.
|Total workload||Contact hours||Credits (ECTS)|
|150 h||60 h||5 CP|
Responsible coordinator of the module PH2028 in the version of SS 2018 was 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.
After participation in the Module the student is able to
- understand the basic mechanisms that lead to patterns and cooperative phenomena in dissipative systems far from the thermodynamic equilibrium
- explain the universal laws leading to pattern formation in reaction-diffusion systems in the bistable excitable and oscillatory regime with prototypical models
- explain the origin of synchronization phenomena in coupled oscillatory networks
- 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
|VO||2||Nonlinear Dynamics and Complex Systems 2||Krischer, K.||
Thu, 10:00–12:00, virtuell
|UE||2||Exercise to Nonlinear Dynamics and Complex Systems 2||
Responsible/Coordination: Krischer, K.
|dates in groups||
Learning and Teaching Methods
The module consists of a lecture and an exercise.
In the thematically structured lecture the learning content is presented. With cross references between different topics the interdisciplinary concepts in nonlinear dynamics are highlighted. In scientific discussions the students are involved to stimulate their analytic-physics intellectual power.
In the exercise the learning content is deepened and exercised using problem examples and simulations. Thus the students are able to explain and apply the learned physics knowledge independently.
black board, powerpoint, videos, script, complementary literature, practise sheets
- Lecture Script
- A.S. Mikhailov, "Foundations of Synergetics I"
- G. Nicolis, "Introduction of Nonlinear Science"
- J. D. Murray "Mathematical Biology II"
- A.S. Mikhailov, G. Ertl, "Chemical Complexity - Self-Organization in Molecular Systems"
Description of exams and course work
There will be an oral exam of 30 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions.
For example an assignment in the exam might be:
- Discuss, how patterns can emerge due to the interplay of reaction and diffusion.
- Illustrate the universal aspect of pattern formation in dissipativ systems with some examples.
In the exam no learning aids are permitted.
Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.
The exam may be repeated at the end of the semester. There is a possibility to take the exam in the following semester.