Mathematical Modelling of Nonlinear Systems
Code  Completion  Credits  Range  Language 

01MMNS  ZK  3  1P+1C  Czech 
 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, 01MAA24, 01LA1, 01LAA2, DIFR, or 01MA1, 01MAB24, 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.Finitedimensional dynamical systems and geometric theory of ordinary differential equations
3.Infinitedimensional 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 dynamicalsystems 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 Lyapunovfunction 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 Lyapunovfunction method in fixedpoint 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, SpringerVerlag, 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:
 Timetable 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  Timetable for summer semester 2024/2025:
 Timetable is not available yet
 The course is a part of the following study plans:

 Aplikované matematickostochastické 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)