Detailed course planning
| Lectures | Exercise classes | Reading | Optional reading | |
| Week 3 |
Lecture 1: Introduction, different type of orbits, rotations on the circle, doubling map. SF2720 lecture 1.pdf Download SF2720 lecture 1.pdf
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Fri 22.1. Section 2.3.3 example of flows on the torus, example 2.10 on page 21, Exercises: 2.1, 2.8, 2.11, 2.14, 2.18, 2.19 |
2.1, 2.2.1, 2.2.2, 2.3.1, 2.4. As preparation for next time: 2.2.3 |
Introduction.pdf Download Introduction.pdf King Oskar II, Poincare and birth of theory of chaotic systems Links to an external site. |
| Week 4 |
More on the doubling map, toral automorphisms, topological transitivity, topological invariance. Intro to topological mixing. |
Fri 29.01 From the handout on Quadratic family Download Quadratic family : Exercise A.5.1, def A.5.1, Exercise A.5.5, Exercise A.5.7 From the book: exercises 3.8, 3.10,
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2.2.3, 3.1, 3.2.1, 3.3, 3.3.1. As preparation for next lecture: 3.3.2 |
Fourier analysis on the torus Download Fourier analysis on the torus Review of metric spaces Download Review of metric spaces
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| Week 5 |
Sensitive dependence on initial conditions, topological mixing, more examples: baker map, more on coding, limit sets. |
Fri 05.02. Homework assignment 1 discussion. |
3.3.2, 7.1.1, 7.1.3, 7.2.1, 7.2.2 |
Box counting dimension of some fractals Links to an external site.
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| Homework assignment 1 is due 05.02.2021 at 8am | ||||
| Week 6 |
Topological entropy: motivation; several definitions and their equivalence; examples. Topological invariance of the topological entropy; some other properties of the topological entropy. |
No exercise class | Sections 3.4.1-3.4.3, 7.1.2, 7.2.1. | |
| Week 7 |
Homeomorphisms of the circle. |
Fri 19.02. Exercises from the book: 3.13-3.15, 3.17; 4.1-4.8. | Section 4.1 | |
| Week 8 |
Diffeomorphisms of the circle. Denjoy's theorem; Denjoy's counterexample. Reading: Denjoy's counterexample Download Reading: Denjoy's counterexample |
Fri 26.02. Exercises from the book: 4.10-4.16 | Section 4.2 | |
| Week 9 |
Maps of the interval; Sharkovsky´s theorem |
Fri 05.03 Homework assignment 2 discussion. | Section 4.3 | |
| Homework assignment 2 is due 05.03.2021 at 8am | ||||
| Week 12 |
Continuous dynamical systems: Poincare section, limit sets for continuous time systems, Poincare-Bendixons theorem. Section 2.3.2, 3.2.2, 4.4 |
Fri 26.03. examples on page 214 of the handout HSD Corollaries 4 and 5 from handout HSD Exercises from handout HSD: 1a, 1d, 2, 4, 10, 12, 15. ChaoticDynamicalSystemsNotes.pdf Download ChaoticDynamicalSystemsNotes.pdf |
Section 2.3.2, 3.2.2, 4.4 Handout HSD: Closed orbits and limit sets from Hirsch-Smale- Devaney book Download Handout HSD: Closed orbits and limit sets from Hirsch-Smale- Devaney book |
History and state of the Hilbert's 16th problem Links to an external site. |
| Week 13 |
Hyperbolic dynamics 1 |
No exercise class this week! | Chapter 5 | |
| Week 15 |
Hyperbolic dynamics 2 |
Fri 16.04 Exercises from the book corresponding to Ch. 5 and 6 | Sec. 6.1 | |
| Week 16 |
Geodesic flows and homogeneous dynamics I Lecture 11.pdf Download Lecture 11.pdf Animation: stable and unstable foliations for the geodesic flow Links to an external site. |
No exercise class | Sec 6.3 |
Geodesic flow on a curved surface Links to an external site.
Short film explaining basics of curved spaces and geodesics Links to an external site. |
| Week 17 |
Geodesic flows and homogeneous dynamics II |
Fri 30.04. Homework assignment 3 discussion. | Sec 6.3 |
Another animation Links to an external site. and an explanation for the above is here: here Links to an external site. |
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Homework assignment 3 is due 30.04.2021 at 8:00 |
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| Week 18 |
Small divisors |
No exercise class | ||
| Week 19 |
Stability of chaotic dynamics SF2720 Lecture 14. Download SF2720 Lecture 14.
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Wed 12.05 old exam problems ChaoticDynamicalSystemsOldExam.pdf Download ChaoticDynamicalSystemsOldExam.pdf |