Nonlinear Dynamics and Complex Systems 1

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.

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)

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.