Dynamical Systems

Dynamical Systems

This module is an introduction to the theory of ordinary differential equations, dynamical systems and their applications. We will cover:

• Basic theory of ordinary differential equations: initial-value problems, Picard-Lindelöf theorem, maximal solutions, linear systems, general solutions;
• Qualitative analysis in mathematical ecology: phase portraits, phase planes, population models, predator-prey and competition dynamics, disease modelling;
• Autonomous systems: phase space, equilibria, periodic, homoclinic and heteroclinic orbits, limit cycles, Hamiltonian systems, orbital mechanics;
• Stability and bifurcation theory: linear and asymptotic stability of equilibria, periodic and homoclinic bifurcations, invariant manifolds, strange and chaotic attractors;

 

Organisational Details

Lectures

Lectures will be on:

• Monday at 10:15-11:45  in Hörsaal IV and
• Wednesday at 10:15-11:45  in Seminar room 9.

Questions/Forum

We will use MS-Teams as forum, there you can ask questions and share your files and notes with each other. You can either join via this link or with the code "q3q4o8r".

Also you can just write me an e-mail at "my-lastname"@math.uni-sb.de or just pass by my office in room 3.03 in building E2 4.

Tutorials

Problem sheets will be released electronically on CMS and solutions should be submitted so that they will be corrected and returned before the tutorial. The problem sheets can be done in groups of up to three people.

Starting on the 31th of october there will be regular tutorials each Friday 14-16 in SR 9 (building E2 4) where the tutor will go through the latest corrected problem sheet. The date of the tutorial will be fixed at the first lecture.

Examination

An oral examination will take place at the end of the semester.

Prerequisites

You should have completed Analysis 1-2 and Linear Algebra 1-2 or equivalent before taking this course.

 

Reading list

You will find a lot of helpfull books in the

• literature list

Another very good source, which you may be able to find on the internet, is

• S. Strogatz, Nonlinear Dynamics and Chaos, CRC Press

 

Other sources

• Jason Bramburger (of Concordia University) has a collection of video lectures on ordinary differential equations and mathematical modelling publicly available on his Youtube channel.
• Also on Youtube, Steve Brunton and Steven Strogatz both have a playlist of videos on dynamical systems as part of lecture courses at the University of Washington and Cornell University, respectively.
   

Note that these courses are not identical to ours, and so they may or may not cover the same topics as us. However, you might find it helpful to see some of the same content presented from different perspectives.