Description:

The course consists of basic terms and results of the theory of finite- and infinitedimensional dynamical systems generated by evolutionary differential equations, and description of bifurcations and chaos. Second part is devoted to the explanation of basic results of the fractal geometry dealing with attractors of such dynamical systems.

Contents:

I. Introductory comments
II. Dynamical systems and chaos
1. Basic definitions and statements
2. Finite-dimensional dynamical systems and geometric theory of ordinary differential equations
3. Infinite-dimensional dynamical systems and geometric theory of ordinary differential equations
4. Bifurcations and chaos; tools of the analysis
III. Mathematical foundations of fractal geometry
1. Examples; relation to the dynamical-systems theory
2. Topological dimension
3. General measure theory
4. Hausdorff dimension
5. Attempts to define a geometrically complex set
6. Iterative function systems
IV. Conclusion - Application in mathematical modelling

Seminar contents:

Exercise makes part of the contents and is devoted to solution of particular examples from geometric theory of differential equations, linearization and Lyapunov-function method, bifurcation analysis and fractal sets.

Recommended literature:

Key references:
[1] F.Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin 1990
[2] M.Holodniok, A.Klíč, M.Kubíček, M.Marek, Methods of analysis of nonlinear dynamical models, Academia, Praha 1986
[3] G.Edgar, Measure, Topology and Fractal Geometry, Springer Verlag, Berlin 1989
[4] K. Falconer, Fractal Geometry - Mathematical Foundations and Applications, J. Wiley and Sons, Chichester, 2014

Recommended references:
[5] D.Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Verlag, Berlin 1981
[6] R.Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, Berlin 1988
[7] G.C. Layek, An Introduction to Dynamical Systems and Chaos, Springer Verlag, Berlin 2015

Media and tools:
Course web page with selected motivation exaamples.

Keywords:

Evolutionary differential equations, dynamical systems, attractors, bifurcations and chaos, topological and Hausdorff dimension, iterative function systems.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar