Nonlinear Dynamics and Complex Systems 2

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 successful completion of the module the students are 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.

PH2027: Nonlinear Dynamics and Complex Systems I (recommended but not essential)

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.

Blackboard, Powerpoint, videos, script, complementary literature, practice sheets.

• Lecture script
• A.S. Mikhailov: Foundations of Synergetics I, Springer Berlin Heidelberg, (2013)
• G. Nicolis: Introduction of Nonlinear Science, Cambridge University Press, (2008)
• J. D. Murray: Mathematical Biology II, Springer, (2011)
• A.S. Mikhailov & G. Ertl: Chemical Complexity - Self-Organization in Molecular Systems, Springer, (2017)

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:

• Diskutieren Sie, wie durch das Zusammenspiel von Reaktion und Diffusion selbstorganisierte Muster entstehen können.
• Verdeutlichen Sie den universellen Aspekt der Musterbildung in dissipativen Systemen anhand von Beispielen.
• What are coupled oscillatory networks?

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.