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Fractals and Chaotic Dynamics - NMNV361

Title:	Fraktály a chaotická dynamika
Guaranteed by:	Department of Numerical Mathematics (32-KNM)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2020
Semester:	winter
E-Credits:	3
Hours per week, examination:	winter s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time
Teaching methods:	full-time

Guarantor:	doc. RNDr. Václav Kučera, Ph.D.
Class:	M Bc. MMIT M Bc. MMIT > Doporučené volitelné M Bc. OM M Bc. OM > Doporučené volitelné M Mgr. NVM M Mgr. NVM > Volitelné
Classification:	Mathematics > Numerical Analysis

Opinion survey results Examination dates WS schedule Noticeboard

Annotation -

Last update: RNDr. Miloslav Vlasák, Ph.D. (10.05.2018)

The course is an introduction to fractal geometry and chaos theory. We will construct the best known types of fractals and derive their basic properties. The key tool will be the concept of iteration. We will focus on iterated function systems (e.g. Barnsley fern), iteration of real functions (Feigenbaum universality) and iteration of complex functions (Mandelbrot and Julia sets). The course is accessible to a wider range of students of mathematics, as well as physics and computer science.

Course completion requirements -

Last update: doc. RNDr. Václav Kučera, Ph.D. (12.05.2018)

The exam is oral. The examination requirements are given by the topics in the syllabus, in the extent to which they they were taught in course.

Literature -

Last update: doc. RNDr. Václav Kučera, Ph.D. (29.10.2019)

Devaney, R.L.: An introduction to chaotic dynamical systems, Westview Press, 2003.

Barnsley, M. F.: Fractals everywhere, Boston: Academic Press Professional, 1993.

Beardon, A.F.: Iteration of rational functions, Graduate Texts in Mathematics vol. 132, Springer, 1991.

Teaching methods - Czech

Last update: doc. RNDr. Václav Kučera, Ph.D. (17.09.2021)

Výuka v akademickém roce 2021/22: Na konkrétním čase konání přednášky se dohodneme před začátkem semestru se studenty podle časových možností všech. V případě distanční výuky budou přednáška konána na platformě ZOOM v čase dle rozvrhu. Materiály k přednášce budou k dispozici na adrese https://su.mff.cuni.cz/ v adresáři home/kucera/Fraktály a chaotická dynamika.

Requirements to the exam -

Last update: doc. RNDr. Václav Kučera, Ph.D. (10.06.2019)

The exam is in oral form. The requirements are given by the scope covered in the lecture.

Syllabus -

Last update: doc. RNDr. Václav Kučera, Ph.D. (12.05.2018)

Fractal geometry: Self-similarity, basic constructions, examples from nature. Hausdorff dimension.

Iterated function systems: Affine self-similar sets, systems of contractions. Existence of the attractor, collage theorem. Algorithms for the generation of attractors, chaos game. Attractor properties.

Iteration of real functions: Bifurcation cascade and diagram. Li-Yorke theorem, Sharkovskii theorem. Quadratic (unimodal) case - definition of chaos, existence of chaotic mappings.

Iteration of complex functions: Quadratic functions, Bernoulli shift, transitivity, sensitivity to initial conditions. Julia and Fatou sets. Examples of the geometry of Julia sets, basic dichotomy. Douady-Hubbard potential, external rays, petals. Mandelbrot set, basic properties, potential, fundamentals of the combinatorics of Mandelbrot's set. Iteration of rational functions, holomorphic dynamics.

Entry requirements -

Last update: RNDr. Miloslav Vlasák, Ph.D. (10.05.2018)

Basic general knowledge of mathematical analysis and linear algebra.