Course: Ordinary Differential Equations 3

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Course title Ordinary Differential Equations 3
Course code KMA/ODR3
Organizational form of instruction Lecture + Exercise
Level of course Master
Year of study not specified
Semester Summer
Number of ECTS credits 3
Language of instruction Czech, English
Status of course Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Course availability The course is available to visiting students
Lecturer(s)
  • Andres Jan, prof. RNDr. dr hab. DSc.
Course content
1.Introductory lecture: Brief summary of basics of the theory of ODEs which are necessary for the course. Historical remarks to the stability theory (program of A. M. Liapunov, the stability theory as a part of the qualitative theory of ODEs, brief information about the recent state of the theory). Types of stability (Liapunov, structural, local dynamical instabilities: deterministic chaos). Theorems about a continuous dependence of solutions of ODEs on initial values. 2.Basic topics of Liapunov stability: Motivating examples for the integrable ODEs. The notion of a stationary (trivial) solution and the transformation of a given solution to trivial solution. Definition of a Liapunov stable solution. 3.Types of Liapunov stability: Definition of a uniformly Liapunov stable solution. Definition of an asymptotically Liapunov stable solution. Definition of a uniformly asymptotically Liapunov stable solution. Definition of a Liapunov unstable solution. Examples of all types of Liapunov stability. 4.Liapunov stability of equilibria of (nonlinear) autonomous systems of ODEs: Definition of stability of equilibria. Definitions of further types of Liapunov stability of equilibria. Illustrative examples. 5.Criteria of Liapunov stability for linear equations with variable coefficients: Lagrange theorem about stability of a trivial solution (necessary and sufficient conditions). Theorem about an asymptotic stability of a trivial solution. 6.Criteria of Liapunov stability for linear equations with constant coefficients: Specification for a particular case of systems with constant coefficients. Routh--Hurwitz criterion (necessary and sufficient conditions). Mikhailov frequency criterion (necessary and sufficient conditions). Illustrative examples. 7.Perturbation theory for linear systems: Investigation of the case, when a matrix with variable coefficients is added to a matrix with constant coefficients. Theorems about the transformation of various sorts of Liapunov stability of a perturbed system to a unperturbed system. Theorem about the investigation of various sorts of stability to the case, when the perturbation is of a polynomial type. 8.Linearization of nonlinear systems: Theorem about Liapunov stability for a system involving a linear autonomous part. Theorem about Liapunov instability for systems involving a linear autonomous part.. Illustrative examples. 9.Second (direct) Liapunov method: The notion of a Liapunov function. Theorem about the Liapunov stability by means of a Liapunov function. Theorem about the Liapunov instability by means of a Liapunov function. 10.Liapunov functions and their construction for further sorts of Liapunov stability: Theorems about the uniform and asymptotic Liapunov stability by means of a Liapunov function. Basins of attraction of a trivial solution. 11.Liapunov functions for autonomous systems: Theorem about the Liapunov stability of a trivial solution to nonlinear systems. Theorem about the Liapunov instability of a trivial solution to nonlinear systems. Theorem about asymptotic Liapunov stability of a trivial solution to nonlinear systems. Illustrative examples. 12.Liapunov functions for the stability of sets: Basics of the Yoshizawa theory of stability of sets. Dissipative systems of ODEs. Criterion of dissipativity of ODEs by means of Liapunov functions (Yoshizawa's theorem). Illustrative examples.

Learning activities and teaching methods
Lecture
  • Attendace - 39 hours per semester
  • Preparation for the Course Credit - 25 hours per semester
  • Homework for Teaching - 25 hours per semester
Learning outcomes
Introduction to basic ideas and principles of the stability theory with practical applications.
Comprehension Understand the stability theory and to apply it in simple examples.
Prerequisites
Students should be familiar with the basic notions of ordinary differential equations to the extent of the course KMA/ODR1.
KMA/ODR2

Assessment methods and criteria
Oral exam

the student has to understand the subject
Recommended literature
  • J. Nagy. (1983). Stabilita řešení obyčejných diferenciálních rovnic. MVŠT, SNTL, Praha.
  • N. P. Bhatia, G. P. Szegö. (2002). Stability Theory of Dynamical Systems. Springer Berlin.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Applied Mathematics (2014) Mathematics courses 1 Summer
Faculty of Science Applications of Mathematics in Economy (2015) Mathematics courses - Summer