Nonlinear Dynamics and Complex Systems 1

Module PH2027

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

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.

  • 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 PH2027 is 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.

Learning Outcome

After participation in the Module the student is familiar with the concepts of nonlinear dynamical systems and
modern techniques to analyse nonlinear ordinary differential equations. This includes to be able to

  1. explain the geometrical approach to dynamical systems and the concepts of stability and bifurcations
  2. perform a phase space analysis envoking dynamical systems tools (programs)
  3. perform a 1- and 2- dimensional bifurcation analysis of a set of coupled ordinary differential equations using continuation software
  4. 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

VU 4 Nonlinear Dynamics and Complex Systems 1 Krischer, K. Freitag, 12:00–14:00
sowie einzelne oder verschobene Termine
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


  • St. H. Strogatz "Nonlinear Dynamics and Chaos"
  • J.M.T. Thompson, H.B. Stewart "Nonlinear Dynamics and Chaos" .
  • E. Ott, "Chaos in Dynamical Systems" 2nd ed.
  • P. Berge, Y. Pomeau, Ch. Vidal, "Order within Chaos"
  • J. D. Murray "Mathematical Biology I"

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.

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.

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

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