News

Registering on LSF

Written on 01.08.23 by Dan Hill

Hi everyone,

if you haven't done it yet, this is just a reminder for you to register for the course on LSF. It makes awarding your grade easier if you've already registered before the exam.

Let me know if you have any issues registering.

All the best,

Dan

Revision Tools

Written on 24.07.23 by Dan Hill

Hi everyone,

First, I just want to note that the `Qualification Sheet' had a typo in it, which has now been corrected. I would recommend you attempting this sheet, as it uses lots of different topics that we covered during the course.

Secondly, here are some additional resources to help with… Read more

Hi everyone,

Secondly, here are some additional resources to help with revision:

The textbook 'Introduction to Applied Nonlinear Dynamical Systems and Chaos' by Stephen Wiggins https://link.springer.com/book/10.1007/b97481 (should be available to download using Uni VPN or on campus) has lots of extra exercises that you can use to practise computing invariant manifolds, etc.
If you are having trouble drawing phase diagrams and bifurcation diagrams, a useful website to help you is
- Phase Plane Plotter : Built with Processing and Processing.js (kevbase.com) - Plotting 2D phase portraits (including in polar coordinates)
- Bifurcation Diagram Plotter : Built with Processing and Processing.js (kevbase.com) - Plotting bifurcation diagrams

I would recommend trying to plot the diagram yourself first, and then comparing with the numerical plot.

If you have any questions during revision, you can send me an email and I will try to help as much as I can.

All the best,

Dan

This week's lectures and the oral exam

Written on 17.07.23 (last change on 19.07.23) by Dan Hill

Hi everyone,

Following a request, I will be going through a summary of the course in Hörsaal IV at the usual time on Friday. The plan is to highlight the important parts of each chapter without going into too much detail. Since I know some people also can't attend this time slot, I will also present the same summary next week on Wednesday 26.07.23 at 12:00-14:00 in Seminarraum 10 (Note, this will unfortunately not be broadcast on MS Teams).

This is also a reminder to send me your choices of dates and time slots for the oral exam by the end of this week, so I can book the room and make sure that an assistant is available. Even if you can't/don't want to do the exam in the first two weeks, please tell me before the end of this week.

Don't forget to register for the exam on the LSF as well.

All the best,

Dan

Registering for the exam

Written on 13.07.23 by Dan Hill

Note that, as well as emailing me your choices of date/time for the oral exam, you should also register for the oral exam on the LSF. Please notify me if you are unable to register.

End of week 14

Written on 12.07.23 by Dan Hill

Updates:

The Lecture notes are now complete, and there is a tutorial on Exercise Sheet 8 tomorrow. We finished the content of the course today, so there will be no lecture on Friday. I am happy to do revision lectures next week in the usual time slots, but I would like you to tell me what I should… Read more

Updates:

Additional Sheets:

There is an additional sheet of questions regarding Chapter 6 on CMS called "Sheet on Chapter 6". If you want to attempt the sheet, you can send your solutions in an email to me and I will provide feedback, but you do not have to.

There is a also a sheet called "Qualification Sheet" on CMS, which is on various topics covered in the course. Those who didn't automatically qualify for the oral exam will have to email solutions to this sheet to me by the end of Thursday of next week (20.07.23 at 23:59). If you automatically qualified for the exam then you do not have to submit solutions but, if you want to, you can also send them in an email to me and I will provide feedback.

Oral exam dates/times

Written on 12.07.23 by Dan Hill

Below are listed dates and times for the oral exam. Please pick three time slots for which you want to do the oral exam and email these to me. My email is hill@math.uni-sb.de

Week 1:

24.07.2023: 10:15, 11:15, 14:15, 15:15

25.07.2023: 10:15, 11:15,

26.07.2023: 10:15, 11:15, 14:15,… Read more

Below are listed dates and times for the oral exam. Please pick three time slots for which you want to do the oral exam and email these to me. My email is hill@math.uni-sb.de

Week 1:

24.07.2023: 10:15, 11:15, 14:15, 15:15

25.07.2023: 10:15, 11:15,

26.07.2023: 10:15, 11:15, 14:15, 15:15

27.07.2023: 10:15, 11:15, 14:15, 15:15

28.07.2023: 10:15, 11:15, 14:15, 15:15

Week 2:

31.07.2023: 10:15, 11:15, 14:15, 15:15

01.08.2023: 10:15, 11:15, 14:15, 15:15

02.08.2023: 10:15, 11:15, 14:15, 15:15

03.08.2023: 10:15, 11:15, 14:15, 15:15

04.08.2023: 10:15, 11:15, 14:15, 15:15

If you cannot/do not want to do the oral exam during these dates, please email me and we can arrange a later date.

Note, I am away for two weeks in August, and so later oral exams will probably have to be in September.

End of week 13

Written on 07.07.23 (last change on 07.07.23) by Dan Hill

Updates:

Lecture notes have been updated with the material covered so far (up to the end of Section 3.2). Next week we will be returning covering strange attractors and fractals. I hope to finish the course on Wednesday, but it depends on how long the material takes.

Exam dates:

The plan… Read more

Updates:

Exam dates:

The plan is to try and complete most of the oral exams in the first two weeks after teaching (so 24.07 - 04.08). I will send out time slots for you to choose next week; the idea is that you would then give me three choices that you would like and I will try to make sure everyone gets at least one of their choices.

If you cannot (or do not want to) do your exam in this two week period, then let me know and we can work out a separate time for the exam to take place. I am away for some of August, so we will have to also work around that.

Exercise Sheets:

The final tutorial will be next Thursday (13.07.23) on the solutions to the eighth exercise sheet.

To automatically qualify for the exam, you must achieve an average of more than 50% across all of the exercise sheets; if you average between 40% and 50%, then you will have to get more than 50% on an additional qualification sheet in order to qualify for the exam. [This extra sheet will be provided once the final exercise sheet has been marked, if any students are between 40% and 50%]

There is an additional sheet of questions regarding Chapter 6 which will be released after we complete the course content. This will be called "Exercise Sheet 9", but does not actually count towards exam qualification. If you want to attempt the sheet, you can send you solutions in an email to me and I will provide feedback, but you do not have to.

End of week 12

Written on 30.06.23 (last change on 07.07.23) by Dan Hill

Updates:

Lecture notes have been updated with the material covered so far (up to Remark 6.9). Next week we will be looking at iterative maps and their connection with chaos.

[Note that most of Section 6.1 (other than the definitions of chaos and attractors) is not examinable, but provides… Read more

Updates:

Lecture notes have been updated with the material covered so far (up to Remark 6.9). Next week we will be looking at iterative maps and their connection with chaos.

[Note that most of Section 6.1 (other than the definitions of chaos and attractors) is not examinable, but provides motivation and context for the rest of the chapter]

Exercise Sheets:

The eighth and final exercise sheet has been released, and is due in one week. There will be a tutorial next Thursday (06.07.23) on the solutions to the seventh exercise sheet.

End of week 11

Written on 23.06.23 (last change on 07.07.23) by Dan Hill

Updates:

Lecture notes have been updated with the material covered so far (up to section 5.3). Next week we will be finish Chapter 5 of the lecture notes (Bifurcations) and begin the sixth and final chapter (Chaos).

Exercise Sheets:

The seventh exercise sheet has been released, and is… Read more

Updates:

Exercise Sheets:

The seventh exercise sheet has been released, and is due in one week. The eighth and final exercise sheet will be released next Friday. There will be a tutorial next Thursday (29.06.23) on the solutions to the sixth exercise sheet.

Exercise Sheet and the exam

Written on 21.06.23 (last change on 21.06.23) by Dan Hill

Hi everyone,

I have been alerted to the fact that the CMS website states that the exam will be an oral exam and so, following university policy, we will be doing oral exams (and not a written exam) at the end of the course. I had forgotten that I had committed to this at the start of the… Read more

Hi everyone,

Also, the current exercise sheet (due this week) on CMS was wrongly stated that is due by the end of Thursday. This is incorrect--- you have until the end of Friday to complete the exercise sheet---and the system has been updated to reflect that.

End of week 10

Written on 16.06.23 (last change on 19.06.23) by Dan Hill

Updates:

Lecture notes have been updated with the material covered so far (end of Chapter 4). Next week we will begin covering Chapter 5 of the lecture notes (Bifurcations).

Room Change:

For next Friday (23.06.23) only, the lecture will be in Hörsaal III.

This is due to a memorial workshop… Read more

Updates:

Lecture notes have been updated with the material covered so far (end of Chapter 4). Next week we will begin covering Chapter 5 of the lecture notes (Bifurcations).

Room Change:

For next Friday (23.06.23) only, the lecture will be in Hörsaal III.

This is due to a memorial workshop happening in the usual room.

Exercise Sheets:

The sixth exercise sheet has been released, and is due in one week. The seventh exercise sheet will be released next Friday. There will be a tutorial next Thursday (22.06.23) on the solutions to the fifth exercise sheet.

Exam:

It looks like there will be enough people qualifying for the exam that I would prefer to give you a written exam rather than doing individual oral exams. Note that this could still change, so let me know if you would prefer to have an oral exam (I won't give some people oral exams and others the written exam, but if enough people want to do an oral exam I might change my mind)

End of week 9

Written on 09.06.23 (last change on 09.06.23) by Dan Hill

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… 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.

End of week 8

Written on 02.06.23 (last change on 02.06.23) by Dan Hill

Updates:

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.

End of week 7

Written on 30.05.23 by Dan Hill

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… 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.

End of week 5

Written on 10.05.23 (last change on 10.05.23) by Dan Hill

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… 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:

Fridays 12.05.23, 19.05.23

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.

End of week 4

Written on 04.05.23 by Dan Hill

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… Read more

Reminder: No lecture tomorrow (05.05.23)! Next lecture is next Wednesday 10.05.23

Updates:

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:

Fridays 05.05.23, 12.05.23, 19.05.23

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.

End of week 3

Written on 28.04.23 (last change on 28.04.23) by Dan Hill

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… 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:

Fridays 05.05.23, 12.05.23, 19.05.23 and Wednesday 17.05.23.

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.

End of week 2

Written on 21.04.23 (last change on 21.04.23) by Dan Hill

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… 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).

Lecture Livestreams

Written on 18.04.23 by Dan Hill

The lectures will be live-streamed via Microsoft Teams, see below for the links:

Wednesday's lecture: link

Friday's lecture: link

End of week 1

Written on 14.04.23 (last change on 28.04.23) by Dan Hill

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… Read more

Updates:

Lecture notes have been updated with the material covered so far (up to Remark 1.24).

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.

Show all

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.