Theoretical nonlinear physics I: Dynamical systems

Contents & literature

Contents of the lecture

This is part I of Theoretical Nonlinear Physics, whose focus are the dynamical systems, i. e., mathematically finite-dimensional systems with purely temporal dynamics. In the following summer semester, part II will follow with a focus on pattern formation in spatially extended systems, i. e. we consider mathematically infinite-dimensional systems with space–time dynamics.

The individual chapters of part I are:

• Introduction
• Experiments and models
• Dynamical systems – distinctions and concepts
• The logistic map
• Flows – linear stability analysis
• Flows – bifurcations and weakly nonlinear analysis
• Deterministic chaos – properties and emergence

Literature

Dynamical systems

• Argyris, J., Faust, G., Haase, M., and Friedrich, R. (2015). An Exploration of Dynamical Systems and Chaos: Completely Revised and Enlarged Second Edition, Springer Berlin Heidelberg.
• Iooss, G. & Joseph, D. (2012). Elementary Stability and Bifurcation Theory, Springer.
• Murray, J. D. (1993). Mathematical Biology, Springer Verlag, Berlin.
• Ott, E. (2002). Chaos in Dynamical Systems, Cambridge University Press.
• Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos, Addison-Wesley.

Nonlinear systems – general

• Ebeling, W. and Feistel, R. (2011). Physik der Selbstorganisation und Evolution. Wiley-VCH Verlag, Weinheim.
• Nicolis, G. (1999). Introduction to nonlinear science. Cambridge University Press.
• Nicolis, G. and Prigogine, I. (1977). Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations. John Wiley & Sons.
• Pismen, L. (2006). Patterns and Interfaces in Dissipative Dynamics (Springer Series in Synergetics). Springer.
• Scott, A. (2006). Encyclopedia of Nonlinear Science. Taylor & Francis.

more literature in the lecture