Mathematical Modelling of Non-linear Systems

Course guarantor:

Michal Beneš

Lecturer:

Michal Beneš

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

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.

Requirements:

Basic course of Calculus, Linear Algebra and Ordinary Diferential Equations, Functional Analysis, Variational Methods (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, DIFR, or 01MA1, 01MAB2-4, 01LA1, 01LAB2, FA1, VAME held at the FNSPE CTU in Prague).

Syllabus of lectures:

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

Syllabus of tutorials:

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.

Study Objective:

Knowledge:

Deterministic dynamical systems, chaotic state description, geometric theory of ordinary and partial differential equations, theoretical fundaments of fractal geometry.

Skills:

Application of linearization method and Lyapunov-function method in fixed-point stability analysis, bifurcation analysis, stability of periodic trajectory, charakteristics of fractal sets and their dimension.

Study materials:

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.

Note:

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Wed
Thu
Fri	roomTR:201 Beneš M. 10:00–11:50 (lecture parallel1) Trojanova 13 horní posluchárna

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikované matematicko-stochastické metody (elective course)
• Fyzika plazmatu a termojaderné fúze (elective course)
• Matematická fyzika (elective course)
• Matematické inženýrství (compulsory course in the program)
• Matematická informatika (elective course)
• Fyzikální elektronika - Počítačová fyzika (elective course)