470-6104/01 – Dynamical Systems and Chaos (DSCH)
| Gurantor department | Department of Applied Mathematics | Credits | 10 |
| Subject guarantor | prof. RNDr. Marek Lampart, Ph.D. | Subject version guarantor | prof. RNDr. Marek Lampart, Ph.D. |
| Study level | postgraduate | Requirement | Choice-compulsory type B |
| Year | | Semester | winter + summer |
| | Study language | Czech |
| Year of introduction | 2021/2022 | Year of cancellation | |
| Intended for the faculties | FS, HGF, FEI, EKF, USP | Intended for study types | Doctoral |
Subject aims expressed by acquired skills and competences
The study of various phenomena has revealed both regular and irregular behaviour, which leads to the need for deep theoretical and applied research. Introduced theories of dynamical systems and chaos in recent decades have penetrated the natural, engineering, and many other areas of human activity. The aim of the course is to respond to the needs of doctoral programs by introducing the necessary theory and practical tools to qualify and quantify dynamic properties.
Teaching methods
Lectures
Individual consultations
Summary
Compulsory literature:
Recommended literature:
Additional study materials
Way of continuous check of knowledge in the course of semester
E-learning
Other requirements
No additional requirements are imposed on the student.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
Block A:
History of chaos and its importance in natural sciences and engineering; Fundamentals of chaos theory (classification of trajectories, parameter influence, nonlinear systems, Poincaré section); Stability, bifurcation (Lyapunov stability, the interface of stable and unstable regions); Analysis of equilibrium states (equilibrium states of continuous DS, basic types of bifurcations); Periodic solutions of dynamical systems (limit cycles, heteroclinic and homoclinic trajectories)
Block B:
Chaotic dynamical systems (bifurcations in chaotic systems, Lyapunov exponents, frequency spectrum); Chaos in discrete and continuous systems (the shift map, transitivity, chaos in the sense of Devaney and Li-Yorke, Poincaré's theorem); Chaos and fractals (fractal, Mandelbrot and Julius sets, IFS and TEA algorithms); Fractals (self-similarity, fractal dimensions and collage theorem)
Block C:
Quantification and qualification tools of dynamic systems (shadowing lemma); Wolf's and Kantz's algorithm for calculation of Lyapunov exponents (Takens' embedding theorem); 0-1 test for chaos, approximate and sampling entropy
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction
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