Geometric Theory of Ordinary Differential Equations

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The seminar consists of the qualitative theory of ODEs dealing with the geometric and topological properties of the solution. In this context, we mention suitably formulated basic results of the existence and uniqueness, continuous dependence on parameters and initial conditions. Main part is devoted to the autonomous systems.

Requirements:

Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01DIFR held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. Basic theorem on the local existence and uniqueness of the solution

2. Theorem of continuous dependence on parameters

3. Differentiability with respect to parameters

4. Continuous dependence on initial conditions, and dfferentiability with respect to initial conditions

5. Basics of the theory of autonomous systems

6. Analysis of solution of autonomous systems (special solutions, phase space)

7. Exponentials of operators

8. Systems 2 x 2

9. Lyapunov stability

10. Limit cycles

11. Poincaré map

12. First integrals and integral manifolds

Syllabus of tutorials:

Study Objective:

Knowledge:

geometric theory of ordinary differential equations, autonomous systems, Lyapunov stability, limit cycles, Poincaré map

Skills:

Formulation of initial value problems for ordinary differential equation. Proving basic mathematical properties of given problems, geometric analysis of the solution.

Study materials:

Key references:

[1] M.W.Hirsch, S.Smale, Differential Equations, Dynamical systems, and Linear Algebra, Academic Press, Boston, 1974

[2] F.Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin 1990

Recommended references:

[3] L. S. Pontryagin, Ordinary Differential Equations. Addison-Wesley, London 1962

[4] D. Schaeffer and J. Cain, Ordinary Differential Equations: Basics and Beyond, Springer-Verlag New York Inc., 2016

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• BS Matematické inženýrství - Matematické modelování (elective course)
• BS Matematické inženýrství - Matematická fyzika (elective course)
• BS Matematické inženýrství - Aplikované matematicko-stochastické metody (elective course)
• BS Informatická fyzika (elective course)
• BS Aplikace softwarového inženýrství (elective course)
• BS Aplikovaná informatika (elective course)
• BS jaderné inženýrství B (elective course)
• BS Jaderné inženýrství C (elective course)
• BS Dozimetrie a aplikace ionizujícího záření (elective course)
• BS Experimentální jaderná a částicová fyzika (elective course)
• BS Inženýrství pevných látek (elective course)
• BS Diagnostika materiálů (elective course)
• BS Fyzika a technika termojaderné fúze (elective course)
• BS Fyzikální elektronika (elective course)
• BS Jaderná chemie (elective course)
• Aplikovaná algebra a analýza (elective course)
• Matematické inženýrství - Matematické modelování (PS)