Nonlinear Dynamics and Complex Systems 1
Module version of WS 2020/1
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|
|WS 2022/3||WS 2021/2||WS 2020/1||WS 2019/20||WS 2017/8||WS 2010/1|
PH2027 is a semester module in English language at Master’s level which is offered in winter 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 PH2027 in the version of WS 2020/1 was Katharina Krischer.
Content, Learning Outcome and Preconditions
This module introduces the basic concepts enabling an understanding of the emergence of cooperative phenomena that are intrinsic to low-dimensional nonlinear systems. The phenomena discussed range from bistable behavior, via sustained oscillations to deterministic chaos. After a historical overview and an introduction into the ideas of nonlinearity and phase space, the lecture follows a classification of dynamical systems according to their phase space dimension, i.e. complexity of the solutions. First, stability and bifurcations of fixed points in one-dimensional systems are discussed. Then oscillations and their emergence in a 2-dimensional phase space are examined. After a discussion of bifurcations of limit cycles and introduction of the Poincare-section and Poincare maps, chaotic dynamics is studied. This includes characterization of chaotic attractors through invariant measures (different dimensions), Lyapunov exponents, routes to chaos and the characterization of experimental, chaotic time series.
Throughout the lecture examples and applications from all fields of natural sciences are discussed, stressing the interdisciplinary aspect of the subject. In the tutorial, the students analyse themselves simple nonlinear equations, applying the techniques introduced in the lecture, and are familiarized with state of the art dynamical systems software.
After participation in the module the students are familiar with the concepts of nonlinear dynamical systems, the differences to linear systems and modern techniques to analyse nonlinear ordinary differential equations. They are able to:
- explain the geometrical approach to dynamical systems and the concepts of stability and bifurcations
- perform a phase space analysis envoking dynamical systems tools (programs)
- perform a 1- and 2- dimensional bifurcation analysis of a set of coupled ordinary differential equations using continuation software
- explain the different routes to low-dimensional deterministic chaos and characterize chaotic dynamics in terms of the most important invariant measures.
No preconditions in addition to the requirements for the Master’s program in Physics.
Courses, Learning and Teaching Methods and Literature
Courses and Schedule
|VO||2||Nonlinear Dynamics and Complex Systems 1||Krischer, K.||
Fri, 12:00–14:00, virtuell
|UE||2||Exercise to Nonlinear Dynamics and Complex Systems 1||
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 universal concepts in nonlinear dynamics are shown. 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 state of the art programs for system analysis. Thus the students are able to explain and apply the learned physics knowledge independently.
Blackboard, powerpoint, videos, textbook, complementary literature, exercises in individual and group workpractise sheets.
- St. H. Strogatz: Nonlinear Dynamics and Chaos, CRC Press, (2000)
- J. Argyris, G. Faust, M. Haase & R. Friedrich: An Exploration of Dynamical Dystems and Chaos, Springer, (2015)
- J.M.T. Thompson & H.B. Stewart: Nonlinear Dynamics and Chaos, Wiley, (2002)
- E. Ott: Chaos in Dynamical Systems, 2nd ed., Cambridge University Press, (2002)
- P. Berge, Y. Pomeau & Ch. Vidal: Order within Chaos: towards a deterministic approach to turbulence, Wiley, (1986)
- J. D. Murray: Mathematical Biology I, Springer, (2007)
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 the qualitative differences of oscillations in linear and nonlinear systems.
- Explain the necessary properties of a dynamical system exhibiting deterministic chaos.
- What is a bifurcation?
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.
Remarks on associated module exams
The exam for this module can be taken together with the exam to the associated follow-up module PH2028: Nonlinear Dynamics and Complex Systems 2 / Nichtlineare Dynamik und komplexe Systeme 2 after the follwoing semester. In this case you need to register for both exams in the following semester.
The exam may be repeated at the end of the semester. There is a possibility to take the exam in the following semester.