News
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09.06.2023
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End of week 9Updates:Lecture notes have been updated with the material covered so far (Linear and Lyapunov stability). Next week, we will begin invariant subspaces and manifolds. Exercise Sheets:The fifth exercise sheet has been released on today and is due on... Read more Updates:Lecture notes have been updated with the material covered so far (Linear and Lyapunov stability). Next week, we will begin invariant subspaces and manifolds. Exercise Sheets:The fifth exercise sheet has been released on today and is due on 16.06.23. The sixth exercise sheet will be released on 16.06.23, due on 23.06.23. |
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02.06.2023
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End of week 8Updates:Lecture notes have been updated with the material covered so far (up to the Chapter 3). Next week, we will begin Chapter 4 (Stability). Exercise Sheets:The fifth exercise sheet will be released on 09.06.23 (next Friday) and is due on 16.06.23. |
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30.05.2023
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End of week 7Updates:Lecture notes have been updated with the material covered so far (end of section 3.4 - Hamiltonian systems). We will start covering Poincare-Bendixson theory this week. Exercise Sheets:The tutorial for the fourth exercise sheet is on Thursday... Read more Updates:Lecture notes have been updated with the material covered so far (end of section 3.4 - Hamiltonian systems). We will start covering Poincare-Bendixson theory this week. Exercise Sheets:The tutorial for the fourth exercise sheet is on Thursday 01.06.23. The fifth exercise sheet will be probably be released on 09.06.23, depending on how much material we have covered by that point. |
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24.05.2023
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Upcoming scheduleHi everyone, I have now returned from the US, and so we will have regular lectures again on Wednesdays and Fridays. However, Leonard (the course tutor) is currently still a little sick, and so we have decided to delay this week's tutorial until next... Read more Hi everyone, I have now returned from the US, and so we will have regular lectures again on Wednesdays and Fridays. However, Leonard (the course tutor) is currently still a little sick, and so we have decided to delay this week's tutorial until next week. I repeat, the tutorial that was meant to happen tomorrow (25.05.23) has been moved to next week (01.06.23). |
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10.05.2023
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End of week 5Reminder: No lecture Friday (12.05.23)! Updates:Lecture notes have been updated with the material covered so far (up to Lemma 3.11 - persistence of linear equilibria). Exercise Sheets:The third exercise sheet is due on Friday 12.05.23. The fourth... Read more Reminder: No lecture Friday (12.05.23)! Updates:Lecture notes have been updated with the material covered so far (up to Lemma 3.11 - persistence of linear equilibria). Exercise Sheets:The third exercise sheet is due on Friday 12.05.23. The fourth exercise sheet will then be released on 12.05.23, to be completed while I am away (due on 19.05.23). Schedule:As mentioned in the previous news post, the following lectures have been cancelled:
Since Thursday 18.05.23 is a public holiday, the lecture on Wednesday 17.05.23 will be replaced with a tutorial in Seminarraum 6 on the solutions to exercise sheet 3. |
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04.05.2023
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End of week 4Reminder: No lecture tomorrow (05.05.23)! Next lecture is next Wednesday 10.05.23 Updates:Lecture notes have been updated with the material covered so far (up to linear phase portraits of autonomous dynamical systems). Next week, we will continue classifying... Read more Reminder: No lecture tomorrow (05.05.23)! Next lecture is next Wednesday 10.05.23 Updates:Lecture notes have been updated with the material covered so far (up to linear phase portraits of autonomous dynamical systems). Next week, we will continue classifying linear phase portraits and look at whether they persist in nonlinear dynamical systems. Exercise Sheets:The third exercise sheet will be released on 05.05.23 (tomorrow) and is due on 12.05.23. The fourth exercise sheet will then be released on 12.05.23, to be completed while I am away (due on 19.05.23). Schedule:As mentioned in the previous news post, the following lectures have been cancelled:
Since Thursday 18.05.23 is a public holiday, the lecture on Wednesday 17.05.23 will be replaced with a tutorial on the solutions to exercise sheet 3. |
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28.04.2023
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End of week 3Updates:Lecture notes have been updated with the material covered so far (up to the end of Section 2.2). Next week, we will finish Chapter 2 and begin Chapter 3 (Autonomous dynamical systems). Exercise Sheets:The second exercise sheet is due by the end of... Read more Updates:Lecture notes have been updated with the material covered so far (up to the end of Section 2.2). Next week, we will finish Chapter 2 and begin Chapter 3 (Autonomous dynamical systems). Exercise Sheets:The second exercise sheet is due by the end of today, the third exercise sheet will be released on 05.05.23 (next Friday) and is due on 12.05.23. Schedule:As mentioned in the previous news post, there are some upcoming lectures that have been cancelled:
Since Thursday 18.05.23 is a public holiday, the lecture on Wednesday 17.05.23 will be replaced with a tutorial on the solutions to exercise sheet 3. |
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27.04.2023
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Upcoming cancelled lecturesUpdates:Some of our upcoming lectures will be cancelled; firstly because of a PhD defence that I need to attend (on 05/05), and secondly because I am in the US for a conference on Applications of Dynamical Systems (from 12/05 to 20/05). As such, there will be no... Read more Updates:Some of our upcoming lectures will be cancelled; firstly because of a PhD defence that I need to attend (on 05/05), and secondly because I am in the US for a conference on Applications of Dynamical Systems (from 12/05 to 20/05). As such, there will be no lectures on:
During this time, you will have some exercise sheets to complete (see below) and there may still be tutorials to attend. Since Thursday 18th May is a public holiday, the tutorial for that week might take place during one of the cancelled lecture slots (to be confirmed).
If you have any questions on the course material while I am away, you can email the course tutor Leonard de Simone at: s8ledesi@stud.uni-saarland.de Exercise Sheets:Exercise sheet 3 will be released on Friday 5th May, to be submitted by the end of the day on Friday 12th May, and exercise sheet 4 will then be released on Friday 12th May, to be completed while I'm gone. |
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21.04.2023
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End of week 2Updates:Lecture notes have been updated with the material covered so far. We finished Chapter 1 in the lecture today, and so we will begin Chapter 2 (on qualitative analysis of dynamical systems) next week. Exercise Sheets:The first exercise sheet is due... Read more Updates:Lecture notes have been updated with the material covered so far. We finished Chapter 1 in the lecture today, and so we will begin Chapter 2 (on qualitative analysis of dynamical systems) next week. Exercise Sheets:The first exercise sheet is due by the end of today, and the second exercise sheet has been released. Solutions to the second exercise sheet are due by the end of next Friday (28.04.2023). |
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19.04.2023
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Typo in Exercise Sheet 1It has been pointed out that there is a typo in Exercise Sheet 1.
In Question 3 (ii), the term xt should be x2
The Exercise Sheet has been updated on the materials page. If you spot any other typos, please let me know (especially if they make the... Read more It has been pointed out that there is a typo in Exercise Sheet 1.
In Question 3 (ii), the term xt should be x2
The Exercise Sheet has been updated on the materials page. If you spot any other typos, please let me know (especially if they make the question impossible!) |
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18.04.2023
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Lecture LivestreamsThe lectures will be live-streamed via Microsoft Teams, see below for the links: Wednesday's lecture: link Friday's lecture: link
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14.04.2023
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End of week 1Updates:Lecture notes have been updated with the material covered so far (up to Remark 1.24). Remember that the proofs in Section 1.1 are not the focus of the course, and so I won't ask you to prove any of the theorems or lemmata regarding the existence and... Read more Updates:Lecture notes have been updated with the material covered so far (up to Remark 1.24). Remember that the proofs in Section 1.1 are not the focus of the course, and so I won't ask you to prove any of the theorems or lemmata regarding the existence and uniqueness of solutions to initial value problems. Room Change:From next week the lectures on Wednesdays will be in Seminarraum 6, and the lectures on Fridays will be in Hörsaal IV. This is so I will be able to stream the lectures on Microsoft Teams. I will make another news about attending remotely next week. Exercise Sheets:The first exercise sheet has been released, and is due in one week. You are allowed to submit in groups of 2, if you wish. Remember that you must average 50% across all exercise sheets to qualify for the exam. Tutorials:Tutorials will be held on Thursdays 16:00-18:00 in Seminarraum 5. They will usually be the week after an exercise sheet is due, beginning on 27.04. |
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
The plan is to hold 25-30 lectures, each of 90 minutes duration.
Lectures will be on Wednesday at 12:00-14:00 in Seminar room 6 and Friday at 12:00-14:00 in Hörsaal IV.
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 two people.
There will be regular tutorials where the tutor will go through the latest corrected problem sheet. Tutorials will be held on Thursday at 16:00-18:00 in Seminar room 5.
Examination
An oral examination will take place at the end of the semester. To qualify for the examination students must have attained an overall score of at least 50% on the problem sheets.
Prerequisites
You should have completed Analysis 1-2 and Linear Algebra 1-2 or equivalent before taking this course.
Reading 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
My lectures might not be the best way for some people to learn, and that's okay! If you still don't understand something after the lecture and would like to hear a second explanation from someone else, I can recommend one of the following:
- Check the books in the Reading list for an explanation
- My colleague 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.