Geometric Theory of Ordinary Differential Equations
Code  Completion  Credits  Range 

01GTDR  Z  2  0+2 
 Course guarantor:
 Michal Beneš
 Lecturer:
 Tutor:
 Michal Beneš
 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, 01MAA24, 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, SpringerVerlag, Berlin 1990
Recommended references:
[3] L. S. Pontryagin, Ordinary Differential Equations. AddisonWesley, London 1962
[4] D. Schaeffer and J. Cain, Ordinary Differential Equations: Basics and Beyond, SpringerVerlag New York Inc., 2016
 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:

 Aplikovaná algebra a analýza (elective course)
 Matematické inženýrství  Matematické modelování (PS)