Stability of Solutions to Non-autonomous Differential Equations (FSI-SN0-A)

Academic year 2026/2027
Supervisor: doc. Ing. Jiří Šremr, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
Aims of the course unit:

Aim of the course: The aim of the course is to acquaint the students with the advanced topics on the theory of stability of solutions to systems of non-autonomous differential equations and to show a possible use of the theoretical results in the analysis of stability of motions (not equilibria only) of mechanical systems with one or more degrees of freedom.

Acquired knowledge and skills: Students will acquire the skills to apply theoretical mathematical apparatus in analysing the stability of the equilibria and the periodic solutions to Hamiltonian systems in classical mechanics. They will thus be ready to study the chaotic behaviour of mechanical systems.
Learning outcomes and competences:
 
Prerequisites:

Linear algebra, differential calculus of functions of one and several variables, integral calculus of functions of one variable, solving of linear ordinary differential equations and their systems, fundamentals of dynamical systems theory (in particular, autonomous systems of ODEs).
Course contents:

The course provides advanced topics on the theory of stability of solutions to systems of non-autonomous differential equations. The stability of solutions will be described in terms of the largest Lyapunov characteristic exponent, a crucial notion for detecting the chaotic behaviour of general dynamical systems.
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:

Attendance at lectures and seminars is obligatory and checked. Absence may be compensated for based on an agreement with the teacher. 

Course-unit credit is awarded on the following conditions: Active participation at seminars.

Examination: The exam tests the knowledge of definitions, theorems, and selected proofs and the ability to apply theoretical apparatus to the given problems. Detailed information will be announced at the end of the semester.
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 2 hrs. compulsory                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture

Lyapunov characteristic exponent
Floquet theory
Stability of systems of linear non-autonomous differential equations
Stability of linear systems with a periodic matrix function
Stability of solutions to systems of non-linear non-autonomous differential equations
Stability of equilibria
Stability of periodic solutions
Stability of solutions to non-autonomous second-order differential equations

    Exercise

Proofs of the statement presented at lectures, examples.
Literature - fundamental:
1.

DEMIDOVICH B. P. Lectures on the mathematical theory of stability. Izdat. "Nauka'', Moscow 1967.
2.

HARTMAN, P. Ordinary differential equations. Philadelphia: SIAM, 2002. ISBN 0-89871-510-5.
3.

YAKUBOVICH V. A., V. M. STARZHINSKII. Linear differential equations with periodic coeffcients, Volume 1. Halsted Press, John Wiley & Sons, New York-Toronto, 1975.
4.

LEONOV G. A., KUZNETSOV N. V. Time-varying linearization and the Perron effects. Int. J. Bifurcation Chaos Appl. Sci. Eng. 17 (2007), 1079–1107.
5.

CHICONE, C. Ordinary Differential Equations with Applications. Netherlands: Springer Nature, 2008. ISBN 9780387226231.
Literature - recommended:
1. HARTMAN, P. Ordinary differential equations. Philadelphia: SIAM, 2002. ISBN 0-89871-510-5.
2.

CHICONE, C. Ordinary Differential Equations with Applications. Netherlands: Springer Nature, 2008. ISBN 9780387226231.
3.

LEONOV G. A., KUZNETSOV N. V. Time-varying linearization and the Perron effects. Int. J. Bifurcation Chaos Appl. Sci. Eng. 17 (2007), 1079–1107.
4.

HARTMAN, P. Ordinary differential equations. Philadelphia: SIAM, 2002. ISBN 0-89871-510-5.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MAI-A full-time study --- no specialisation -- Cr,Ex 0 Elective 2 2 W
N-MAI-P full-time study --- no specialisation -- Cr,Ex 0 Elective 2 2 W