Qualitative Theory of Ordinary Differential Equations
General data
Course ID: | 1000-135RRJ |
Erasmus code / ISCED: |
11.133
|
Course title: | Qualitative Theory of Ordinary Differential Equations |
Name in Polish: | Jakościowa teoria równań różniczkowych zwyczajnych |
Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
Course groups: |
(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics |
ECTS credit allocation (and other scores): |
6.00
|
Language: | English |
Type of course: | elective courses |
Prerequisites (description): | (in Polish) 1000-114aRRZa lub 1000-114aRRZb |
Short description: |
We will address the limit behaviour of trajectories of an ordinary differential equation, its invariant sets and the approach to a differential equation seen as a dynamical system. |
Full description: |
1. Lapunov stability and asymptotic stability. 2. Neighborhood of an equilibrium point. Hadamard-Peron Theorem and Grobman-Hartman Theorem. 3. Periodic trajectories and limit cycles. Poincare-Bendixon Theorem and Dulac Theorem. 4. Phase portraits of vector fields in the plane. 5. Elements of bifurcation theory. Saddle-node bifurcation, Anronov-Hopf bifurcation and period doubling bifurcation. 6. Equations with a small parameter. Perturbations to Hamilton system: limit cycle generation in case of one degree of freedom, information about the KAM theory. Relaxing oscillations. 7. Chaos and attractors. |
Bibliography: |
V.I.Arnold, Ordinary Differential Equations V.I.Arnold, Theory of Differential Equations J.Hale, Ordinary Differential Equations, Krieger, 1980. A.A. Andronov et al., Qualitative theory of second order dynamical systems. John Wiley and Sons, 1973 (oryg. ros. Nauka, Moskwa 1966). A.A. Andronov et al., Theory of bifurcations of dynamical systems on a plane. John Wiley and Sons, 1973 (oryg. ros. Nauka, Moskwa 1967). D.K. Arrowsmith and C.M. Place, Theory of bifurcations of dynamical systems on a plane. Chapman and Hall, 1982. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos. Springer-Verlag, 1990. R.L. Devaney, An introduction to chaotic dynamical systems. Cummings, 1986. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag 1983. |
Learning outcomes: |
Introduction to the qualitative analysis of the ordinary differential equations and with introductory notions of the dynamical systems theory |
Assessment methods and assessment criteria: |
Written and oral exam |
Classes in period "Winter semester 2023/24" (past)
Time span: | 2023-10-01 - 2024-01-28 |
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MO WYK
CW
TU W TH FR |
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Henryk Żołądek | |
Group instructors: | Henryk Żołądek | |
Students list: | (inaccessible to you) | |
Examination: |
Course -
Examination
Lecture - Examination |
Classes in period "Winter semester 2024/25" (future)
Time span: | 2024-10-01 - 2025-01-26 |
Navigate to timetable
MO TU W TH FR |
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Henryk Żołądek | |
Group instructors: | Henryk Żołądek | |
Students list: | (inaccessible to you) | |
Examination: |
Course -
Examination
Lecture - Examination |
Copyright by University of Warsaw.