Theoretical nonlinear physics II: Spatially extended systems and pattern formation

Contents & literature

Contents of the lecture

1. Introduction
2. Repetition of dynamical systems [attractors and their stability, local bifurcations and their normal forms]
3. Systems and models – from transport equations to order parameter equations [concept of von balance and transport equations, selected physical applications, basic model equations of structure formation]
4. Stationary states in spatially extended systems and their linear stability [Turing instability, Hopf instability, codimension 2 instabilities, classification of instabilities of one-dimensional patterns]
5. Bifurcations and weakly nonlinear theory
6. Secondary instabilities, defects and chaos [space–time chaos in structure formation]

In the course of the lectures, the topics

• Reduction methods for order parameter equations
• Multi-scale analysis
• Galerkin approximation
• Role of symmetries
• Selection mechanisms for patterns
• Models such as Swift-Hohenberg/Ginzburg-Landau/Kuramoto-Sivashinsky/reaction-diffusion equations

will also be mentioned.

Literature

Chapter 2:

• Argyris, J.; Faust, G.; Haase, M. & Friedrich, R., An Exploration of Dynamical Systems and Chaos: Completely Revised and Enlarged Second Edition, Springer Berlin Heidelberg, 2015
• Strogatz, S. H., Nonlinear Dynamics and Chaos Addison-Wesley, 1994

Chapters 3–6:

• G. Nicolis, Introduction to nonlinear science, Cambridge University Press, 1999
• M. Bestehorn, Hydrodynamik und Strukturbildung: Mit einer kurzen Einfuhrung in die Kontinuumsmechanik (Springer-Lehrbuch) (German Edition) Springer, 2006
• R. Hoyle: Pattern formation, Cambridge Univ. Press 2007
• P. Colinet, J. C. Legros & M. G. Velarde, Surface-Tension-Driven Instabilities, Wiley-VCH, 2001
• M.C. Cross, H. Greenside: Pattern formation and dyanmics in nonequilibrium systems, Cambridge Univ. Press 2009
• M.C. Cross , P.C. Hohenberg, Pattern formation outside equilibrium, Rev. Mod. Phys. 65, 854 (1993)
• A.C. Newell, T. Passot, J. Lega, Order parameter equations for pattern, Annu. Rev. Fluid Mech. 25, 399 (1993)
• D. Walgraef: Spatio-temporal pattern formation, Springer 1997

more literature in the lecture