Dynamical systems and nonlinear differential equations


VO 250115-1 (5 ETCS) and PS 250010-1 (2 ECTS)

Lecturer: Prof. Henk Bruin

Email H. Bruin for further information for this course.

Proseminar by: Prof. Henk Bruin

Announcements

For Friday 1 March the lecture will be in HS8 at Oskar Morgensternplatz 1.
Please note also the change of lecture rooms for the rest of the semester.

The proseminar will be every other Friday, starting on March 8.


Rector's Day is March 12
Easter Break is from March 25 to April 7
Pentacost is on May 19 & 20 (Vorlesungsfrei).


Schedule



Day Time Roomfromto
Tuesday 16:45-18:15 SR8 Kolingasse2.3.202427.06.2021
Friday
Lecture (odd weeks)
11:30-13:00 SR719.3.202430.06.2021
Friday
Proseminar (even weeks)
11:30-13:00 SR719.3.202430.06.2021

Weekly Progress

Day Topic Material
Week 1
1/3/2024 Introduction, 1-D ODE
Stationary points and their stability
simple polulation growth models
Notes of lecture 1
Lecture on a population dynamics model by Anima Nagar
Week 2
5/3/2024 Quadratic family
Cobweb diagram
Feigenbaum diagram
Notes of lecture 2.

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's constant is not part of the course material.)
8/3/2024 Proseminar Exercise 1-5 of the Exercise sheet. The checkmark list on Moodle.
Week 3
12/3/2024 Rektorstag -- no class
15/3/2024 Class canceled
Week 4
19/3/2024 Bifurcations in 1D
22/3/2024 Proseminar Exercise 6,7,8,9 and 11 of the Exercise sheet. The checkmark list on Moodle.
Week 5
9/4/2024 Mathematical Chaos
Topological transitivity
Notes of today's lecture
12/4/2024 Schwarzian Derivative Notes of today's lecture
Week 6
16/4/2024 Hartman-Grobman Notes of today's lecture

Online lecture by Strogatz
19/4/2024 Proseminar Exercise 11, 16, 17 and 18 of the Exercise sheet and also prove the following properties of the Schwarzian derivative (see also the lecture notes of April 12):
1. S(f o g) = (Sf) o g . (g')2 - Sg.
2. Sf = 0 if and only if f is a Möbius transformation.
Hint: Show that Sf = -2√(f') . (1/√(f'))''. Then Sf = 0 is equivalent to (1/√(f'))'' = 0, and that is easier to solve.
The checkmark list on Moodle.
Week 7
23/4/2024 Lotka-Volterra (predator-prey) systems
Lyapunov functions.
Notes of today's lecture
Lecture by Strogatz.
26/4/2024 Limit Cycles
Van der Pol system
ω-limit and α-limit sets
Week 8
30/4/2024 ω-limit and α-limit sets
Poincaré-Bendixson Theorem
19/4/2024 Proseminar Exercise 17 and 18 of the Exercise sheet and also prove the following properties of the Schwarzian derivative (see also the lecture notes of April 12):
1. S(f o g) = (Sf) o g . (g')2 - Sg.
2. Sf = 0 if and only if f is a Möbius transformation.
Hint: Show that Sf = -2√(f') . (1/√(f'))''. Then Sf = 0 is equivalent to (1/√(f'))'' = 0, and that is easier to solve.
The checkmark list on Moodle.
Week 9
7/5/2024 Circle maps
Rotation numbers
Notes of the today's lecture
10/5/2024 Rotation numbers
Symbolic dynamics
Week 10
14/5/2024 Smale's horseshoe
More symbolic dynamics
Notes of the today's lecture
Nagar's Lecture 12 on dynamics of the horseshoe attractor.
17/5/2024 Proseminar Exercise 18, 21 and 22 of the Exercise sheet and also prove the following properties of the Schwarzian derivative (see also the lecture notes of April 12):
1. S(f o g) = (Sf) o g . (g')2 - Sg.
2. Sf = 0 if and only if f is a Möbius transformation.
Hint: Show that Sf = -2√(f') . (1/√(f'))''. Then Sf = 0 is equivalent to (1/√(f'))'' = 0, and that is easier to solve.
The checkmark list on Moodle.
Week 11
21/5/2024 Pentacost, no class
24/5/2024 Toral automorphisms Lecture by Anima Nagar: Lectures 33 and
Lecture 34 on chaos in hyperbolic toral automorphisms.
Week 12 Topological Entropy Notes on topological entropy (Only Sections 1.3 and 1.4 are course material)
Several lectures by Anima Nagar on entropy:
Lecture 27 on measuring chaos, topological entropy.
Lecture 28 on Adler's et al. definition of topological entropy.
Lecture 29 on Bowen's definition of topological entropy.
Lecture 30 on equivalence of the two definitions of topological entropy.
28/5/2024
31/5/2024 Proseminar Exercise 41, 42, 43, 46 and 48 of the Exercise sheet
Week 13
4/6/2024 More on Topological Entropy
Notes on topological entropy
(Only Sections 1.3 and 1.4 are course material)
7/6/2024 Oscillators and resonance
driven, coupled oscialltors
Notes of today's lecture
Week 14
11/6/2024 The cusp bifurcation
hysteresis
Notes of today's lecture
Lecture by Strogatz)
14/6/2024 Proseminar Exercises 40 and 48 of the Exercise sheet Furthermore:
1. Let A = {1,2,...,d} be an alphabet and Σ = AN. Let (X = AN,σ) be a subshift (i.e., X is a subset of Σ) with metric d(x,y) = 2 -n for n = min(k : x k ≠ y k ). Let p(n) = #{subwords of length n in X}.
a) Show that htop(σ) = limn (log p(n))/n.
b) What is the entropy of the full shift on d letters?
c) Compute the entropy of a subshift of finite type with transition matrix A = (2 1 // 1 1)

2. Let Rα(x) = x+α mod 1 be an irrational rotation on the unit circle. Define symbolic dynamics based on the partition [0, α) with symbol 0 and [α, 1) with symbol 1. Let i:S1 → {0,1}N be the itinerary map.
a) An n-cylinder Z[x] is defined as the set {y : i(y) k = i(x) k for 0 ≤ k < n}. What do n-cylinders look like as subset of the circle?
b) How many n-cylinders does a circle rotation have? Derive the entropy of Rα from this.
Week 15
18/6/2024 No lecture
21/6/2024 No lecture
Week 16
25/6/2024 The Lorenz system
Notes of today's lecture
Lectures by Strogatz on the Lorenz system and another one.
28/6/2024 Proseminar Exercises 38, 51, 53, 55 of the Exercise sheet

References/Background Reading

Assessment

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 2024