Dynamical systems and nonlinear differential equations

VO 250010-0 (5ECTS) and PS 250010-1 (2ECTS)

Lecturer: Prof. Henk Bruin

Email H. Bruin for further information for this course.

Proseminar by: Prof. Henk Bruin


Due to Covid measures, you will have to register at this course on u-space.

The last class was on June 15 2021. The last proseminar will be on June 18 2021.

As agreed during the lecture, the exam will be oral. You can make an appointment by sending me an email.

Rector's Day is March 12
Easter Break is from March 29 to April 9
Pentacost is on May 24 & 25 (Vorlesungsfrei).


In the below schedule, the location HS13 might actually be replace by online lectures, subject to the current Covid19-Measures.

Day Time Roomfromto
Tuesday 15:00-16:30 HS132.3.202129.06.2021
Friday 9:45-10:30HS135.3.202125.06.2021
10:30-11:15 HS135.3.202125.06.2021

Contents of the course

Basic notions for continuous and discrete dynamical systems; flows, attractors, and chaos; stability of stationary points by linearization and by Lyapunov functions, Poincaré-Bendixson theory, bifurcations.

Weekly Progress

Notes of lecture 6
Day Material
Week 1
2/3/2021 Introduction, 1-D ODE
Stationary points and their stability
simple polulation growth models
Notes of lecture 1
Lecture on a population dynamics model by Nagar.
5/3/2021 Quadratic family
Cobweb diagram
Feigenbaum diagram
Notes of lecture 2 (with typos corrected).
The link of recording is sent to you in the email of March 5.
Applets used in the lecture: cobweb diagram, the bifurcation diagram and another one , all for the logistic family.
Lecture by Strogatz
A video on the Feigenbaum diagram. (Feigenbaum constant are not part of the course material.)
Week 2
9/3/2021 Bifurcations in 1D
(transcritical, saddlenode/fold, pitchfork)
12/3/2021 No class (Rektorstag)
Week 3
16/3/2021 Sensitive dependence on initial conditions
Mathematical chaos (in the sense of Devaney)
Lyapunov exponents
Notes of lecture 3
Lecture on sensitive dependence on initial conditions by Nagar.
Lecture on Devaney chaos by Nagar.
19/3/2021 Proseminar Exercises 1-5
Week 4
23/3/2021 Linear systems
Structural stability
Hartman-Grobman Theorem

Lecture by Strogatz
Exercise + solution on Structural stability
27/3/2021 Lotka-Volterra Predator-Prey model
Lyapunov functions

Lecture by Strogatz
Week 5 This is after the Easter break
13/4/2021 Limit cykels
Van der Pol equation
Hopf bifurcation

Lecture by Strogatz
16/4/2021 Proseminar Exercises 6-10
Week 6
20/4/2021 Symbolic dynamics and Markov partitions
Sharkovskii's Theorem
Notes of the lecture on symbolic dynamics
Lecture 23 on introduction to symbolic dynamics.
Nagar's Lecture 17 on transitivity and Devaney chaos and Sharkovskiy's Theorem.
Some typed notes on Symbolic dynamics
23/4/2021 Smale's Horseshoe
Notes of the lecture on Smale's horse-shoe
Lecture 12 on dynamics of the horseshoe attractor.
Week 7
27/4/2021 Poincaré-Bendixson Theorem Notes of the Poincaré-Bendixson Theorem
Lecture by Strogatz
30/4/2021 Proseminar Exercises 11-13
Week 8
4/5/2021 Circle homeomorphisms
rotation numbers
Notes of the lecture on circle homeomorphisms
7/5/2021 canceled
Week 9
11/5/2021 The cusp bifurcation
Notes of the cusp bifurcation
Recorded lecture on the cusp bfurcation will be posted here.
Lecture by Strogatz
14/5/2021 canceled
Week 10
21/5/2021 Proseminar: Exercises 13-16.
Week 11
25/5/2021 No class
28/5/2021 Oscillators Notes of the lecture on oscillators
Lecture by Wyslouch
Week 12
1/6/2021 Hamiltonian dynamics
Euler-Lagrange equations
Notes of the lecture on Hamiltonian dynamics
Online lecture by Strogatz
4/6/2021 Proseminar Exercises 15-18.
Week 13
8/6/2021 Noether's Theorem
n-body problem.
Notes of the lecture on Noether's Theorem
Further text on Hamiltonian dynamics and the n-body problem
11/6/2021 Kepler's Problem
2-body problem
Week 14
15/6/2021 Lorenz attractors
Lectures by Strogatz on the Lorenz system and another one.
Notes of the lecture on the Lorenz system
18/6/2021 Proseminar Exercises 19-22

The course will be given in English

Contents of the course (B = Exercises , S = Schmeiser's notes, T = Teschl's book), St = Strogatz lecture on youtube.

References/Background Reading


Assessment of the course will be by oral exam, as discussed in class. The default language is English, but German is possible too.

Course material (Hand-outs)

Updated February 26 2021