Physical Modelling of Complex Systems

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About the course

This 6 ECTS course is taught by Enrico Carlon and Lendert Gelens and deals with the modelling of complex systems using different mathematical models and with various applications, mostly centered around biophysics, but no knowledge about biology is required at all and the course is still interesting even if students have no specific interest in biological applications. Topics include: population growth, dynamical systems and phase portraits, quasi steady state approximation, motifs, non-linear oscillators and limit cycles, chaos theory, synchronization, travelling waves and (Turing) pattern formation: see below.

The evaluation is based on exercises that should be made during the year, but which don't have any strict deadlines and you are free to submit them whenever you like. Exercises include some analytic calculations (not too difficult) but are more focused on numerical or graphical work, using whatever program you like. The exam consists of both professors asking questions specifically related to the exercises you made, but also about the theory seen during the lectures which may not have been used directly in the exercises, so it is advised to fresh up on the lectures even after making all assignments. Both professors are very kind and nudge you towards the final answer, and will also tell you when you're correct. In 2021, the exam had a duration of 1 hour (30 mins for each professor).

Exercises

The exercises can be found here: Media:Physical modelling problem set.pdf. Exercises coloured in yellow had to be made and submitted before the exam for academic year 2020-2021. There was an additional exercise on pattern formation not included in this PDF, which can be found here: Media:pattern formation exercise.png, the same exercise including some first steps and solutions (without much detail though) is also given here: http://homepages.vub.ac.be/~sdebuyl/Turing_course_red.pdf.

Exam

21 May 2021

Additional questions from Carlon: Compare Gompertz law and logistic equation around the fixed point. Explain difference between coherent and incoherent feed forward loops. Why do we observe a lot of certain FFL but not many of another type (situation where protein X activates Y but represses Z is not easily realised in Nature, e.g.)? When do we have oscillations in 2D systems (when we have a negative feedback loop and autocatalysis)? What is the advantage of having an incoherent FFL?

Additional questions from Gelens: discuss the waterwheel equations (parameters) and draw the bifurcation diagram of the Lorenz system and discuss it (subcritical Hopf bifurcation,...). Discuss the Adler system (sketch a few cases for varying parameters). What is the generalisation for a whole population of oscillators and how did we analyse this (Kuramoto model: by introducing an order parameter: equations became uncoupled and related to Adler equation)? Additional questions about the Allee effect in travelling waves, discuss this, what if the parameter a is increased?