Ruprecht Karls Universität Heidelberg

Non-linear dynamics

Non-linear dynamics is the study of dynamical processes that are governed by deterministic but non-linear laws. From the mathematical point of view, we deal with systems of ordinary differential equations (ODEs). Due to the non-linearity, 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 non-linear 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.

Non-linear dynamics can be studied through non-linear 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 non-linear systems includes negative feedback, homeostasis, positive feedback, bistability, switch-like behaviour and oscillations, which occur in many natural and man-made 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 reaction-diffusion 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



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 Edelstein-Keshet, 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
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