MU24055 Qualitative Methods for Ordinary Differential Equations

Mathematical Institute in Opava
Winter 2020
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Jana Kopfová, Ph.D. (lecturer)
Guaranteed by
RNDr. Jana Hantáková, Ph.D.
Mathematical Institute in Opava
Timetable
Wed 11:25–13:00 108
  • Timetable of Seminar Groups:
MU24055/01: Wed 15:35–17:10 108, J. Kopfová
Prerequisites (in Czech)
TYP_STUDIA ( N )
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
This is an advanced course of the theory of ordinary differential equations. The global and qualitative characteristics of ordinary differential equations and their systems will be investigated. The main focus will be on the asymptotic behavior of the different types of solutions and their stability.
Syllabus
  • 1. Qualitative analysis of ordinary first-order differential equations (monotonicity and convexity of solutions, Barrow's formula)
    2. Stability (Lyapunov function, Lyapunov exponent, stability of the linearized system)
    3. Geometry of solutions of autonomous systems of differential equations in the plane (vector field index relative to the Jordan curve, Umlaufsatz, methods for finding cyclical solutions)
    4. Geometry of solutions of autonomous systems of differential equations in the 3-dimensional space (Lorenz's strange attractor)
    5. Qualitative analysis of solutions of non-autonomous systems of differential equations (non-unique solutions, the equation of pendulum)
Literature
    required literature
  • G. Teschl. Ordinary differential equations and dynamical systems. Providence, 2012. info
  • P. Hartman. Ordinary Differential Equations. Baltimore, 1973. info
    recommended literature
  • T. Bárta, D. Pražák. OBYČEJNÉ DIFERENCIÁLNÍ ROVNICE, sbírka úloh a řešených příkladů. URL info
  • J. Andres, J. Fišer. Obyčejné diferenciální rovnice 3: úvod do teorie stability. Olomouc, 2015. URL info
  • L. S. Pontrjagin. Ordinary differential equations. Massachusetts, 1696. info
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course can also be completed outside the examination period.
Teacher's information
To obtain the course credit prior to examination, at least 3 problems are to be solved (a solution of a problem during the exercise or a homework). The exam is oral.

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