Email H. Bruin for further information for this course.
Day | Time | Room | from | to |
---|---|---|---|---|
Tuesday | 16:45-18:15 | SR8 Kolingasse | 2.3.2024 | 27.06.2021 |
Friday Lecture (odd weeks) | 11:30-13:00 | SR7 | 19.3.2024 | 30.06.2021 |
Friday Proseminar (even weeks) | 11:30-13:00 | SR7 | 19.3.2024 | 30.06.2021 |
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 = A^{N}. Let (X = A^{N},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 h_{top}(σ) = lim_{n} (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 R_{α}(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:S^{1} → {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 R_{α} 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 |
Assessment of the course will be by oral exam, as discussed in class.
The default language is English, but German is possible too.
Smale's horseshoe | A homoclinic tangle | Another homoclinic tangle |