470-6104/01 – Dynamical Systems and Chaos (DSCH)

Gurantor departmentDepartment of Applied MathematicsCredits10
Subject guarantorprof. RNDr. Marek Lampart, Ph.D.Subject version guarantorprof. RNDr. Marek Lampart, Ph.D.
Study levelpostgraduateRequirementChoice-compulsory type B
YearSemesterwinter + summer
Study languageCzech
Year of introduction2021/2022Year of cancellation
Intended for the facultiesFEI, FS, EKF, USP, HGFIntended for study typesDoctoral
Instruction secured by
LoginNameTuitorTeacher giving lectures
LAM05 prof. RNDr. Marek Lampart, Ph.D.
MAL0337 doc. RNDr. Michal Málek, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Examination 2+0
Part-time Examination 12+0

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:

[1] P. Blanchard, RL. Devaney, G.L Hall, Differential Equations, Brooks/Cole, Cengage Learning, 2011. ISBN-13:978-0-495-56198-9 [2] J.T. Sandefur, Discrete dynamical systems, Theory and Applications, Clarendon Press, Oxford, 1990. ISBN 0-19-853384-5

Recommended literature:

[1] F.R. Marotto, Mathematical Modeling Using Discrete Dynamical Systems, Brooks/Cole, USA. ISBN 0-495-01417-6 [2] R.L. Devaney, An Introduction to Dynamical Systems, Westview Press, Colorado, 2003. ISBN-13:978-0-8133-4085-2

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

Part-time form (validity from: 2021/2022 Winter semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of pointsMax. počet pokusů
Examination Examination   3
Mandatory attendence participation:

Show history

Conditions for subject completion and attendance at the exercises within ISP:

Show history

Occurrence in study plans

Academic yearProgrammeBranch/spec.Spec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2024/2025 (P0541D170005) Computational and Applied Mathematics K Czech Ostrava Choice-compulsory type B study plan
2024/2025 (P0541D170005) Computational and Applied Mathematics P Czech Ostrava Choice-compulsory type B study plan
2023/2024 (P0541D170005) Computational and Applied Mathematics P Czech Ostrava Choice-compulsory type B study plan
2023/2024 (P0541D170005) Computational and Applied Mathematics K Czech Ostrava Choice-compulsory type B study plan
2022/2023 (P0541D170005) Computational and Applied Mathematics K Czech Ostrava Choice-compulsory type B study plan
2022/2023 (P0541D170005) Computational and Applied Mathematics P Czech Ostrava Choice-compulsory type B study plan

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner

Assessment of instruction

Předmět neobsahuje žádné hodnocení.