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). 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). 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 two 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 action potentials and spikes, 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. A very modern example is the Min-system, which is a spatiotemporal oscillator in bacteria that can also be reconstituted in the test tube.

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 - 11.00 am in seminar room A of the Mathematikon (INF 205). It has been moved there from kHS at Philosophenweg 12 due to better accessibility. Every two weeks on Wednesday afternoons the solutions to the exercises will be discussed in a tutorial. If you attend the course and solve more than 60 percent of the exercises, you earn 4 credit points. We recommend to complement this course by the one on stochastic dynamics (Monday 2.15 - 4.00 pm at INF 308, tutorial in the complementary weeks).

Exercises

Solutions are to be handed in at the lecture one week after assignment. You can work in groups of two if you want to.

• 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