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Course, academic year 2018/2019
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Fractals and Chaotic Dynamics - NMNV361
Title in English: Fraktály a chaotická dynamika
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. RNDr. Václav Kučera, Ph.D.
Class: M Bc. MMIB
M Bc. MMIB > Doporučené volitelné
M Bc. OM
M Bc. OM > Doporučené volitelné
M Mgr. NVM
M Mgr. NVM > Volitelné
Classification: Mathematics > Numerical Analysis
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 - Czech
Last update: doc. RNDr. Václav Kučera, Ph.D. (12.05.2018)

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.

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.

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