
Nonlinear dynamics
Nonlinear dynamics is the study of dynamical processes that are governed by deterministic but nonlinear laws. From the mathematical point of view, we deal with systems of ordinary differential equations (ODEs). Due to the nonlinearity, very interesting phenomena occur. Most importantly, the superposition principle valid for linear systems does not hold. On the one hand, this means that small perturbations can decay again, which is an important prerequisite to obtain stable limit cycles (oscillations, minimal dimension 2). On the other hand, small perturbations need not to stay small, thus small variations in initial conditions can lead to very different results (deterministic chaos, minimal dimension 3). One very important aspect of nonlinear dynamics are bifurcations, when the solutions to the corresponding system of ODEs suddenly changes its character as some parameter goes through a critical value. In physics, such situations occur for example at phase transitions.
Nonlinear dynamics can be studied through nonlinear differential or difference equations and in both cases, graphical methods are very helpful. In 2 dimensions, one can use phase plane analysis. The range of typical behaviour of nonlinear systems includes negative feedback, homeostasis, positive feedback, bistability, switchlike behaviour and oscillations, which occur in many natural and manmade systems.
This course offers an introduction to the mathematical and computational tools needed to understand these systems properties. We also will discuss applications in biophysics, including molecular processes like enzyme kinetics, cellular processes like hearing or spiking, and evolutionary processes like coexistence of competing species. At the end of the course, we will also discuss the extension to pattern formation, which means that we also include space. Then we deal with partial differential equations (PDEs) rather than with ODEs. One famous example is the Turing instability, where a reactiondiffusion system spontaneously develops a stripe pattern.
The course is designed for physics students in advanced bachelor and beginning master semesters (students from other disciplines are also welcome). It will be given in English. A basic understanding of physics and differential equations is sufficient to attend. The course takes place every Wednesday from 9.15  10.45 am in room 106 at Philosophenweg 12. Every two weeks on Wednesday from 2.15  3.45 pm the solutions to the exercises will be discussed in a tutorial (seminar room 2.403 in KIP, INF 227, starting October 30). If you attend the course and solve more than 50 percent of the exercises, you earn 4 credit points. We recommend to complement this course by the one on stochastic dynamics (Monday 2.15  3.45 pm at KIP, tutorial in the complementary weeks). The last lecture will take place on Jan 29.
Material for the course
 Introduction Oct 16 2013
 Presentation on pattern formation in biological systems Jan 29 2014
 Script written by Maria Heinz (final version Jan 29 2014)
Exercises
 1st set Oct 23 2013
 2nd set Nov 6 2013
 3rd set Nov 20 2013
 4th set Dec 4 2013
 5th and last set Dec 20 2013
Software
 pplane, a great phase plane analysis tool for Matlab from John Polking at Rice University
 xppaut, a great standalone phase plane analysis and bifurcation tool from Bard Ermentrout at the University of Pittsburgh
 Here you find information on dynamic systems tools in Mathematica
 Matlab program for the glycolysis oscillator
 Matlab program for the van der Pol oscillator
 Matlab program for a simple model for epidemics
Recommended reading
 SH Strogatz, Nonlinear dynamics and chaos, Westview 1994
 M Cross and H Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press 2009
 JD Murray, Mathematical biology, 3rd edition (now in volumes I and II), Springer 2002>
Additional reading in biophysics
 R. Phillips and coworkers, Physical Biology of the Cell, 2nd edition Garland Sci. 2012
 L EdelsteinKeshet, Mathematical Models in Biology, Random House 1988
 J Keener and J Sneyd, Mathematical Physiology, Springer 1998
 C Fall et al, eds, Computational Cell Biology, Springer 2002
 M Nowak, Evolutionary Dynamics, Harvard University Press 2006
 U Alon, An Introduction to systems biology, Chapman and Hall/CRC 2007