Code: 01DRCH |
Differential Equations and Chaos |
Lecturer: prof. Dr. Ing. Michal Beneš |
Weekly load: 0+2 |
Completion: A |
Department: 14101 |
Credits: 2 |
Semester: W |
- Description:
-
Basic theorem on the local existence and uniqueness of the solution. Continuous dependence and differentiability of the solution. Basics of the theory of autonomous systems. Analysis of solution of autonomous systems (special solutions, phase space). Exponentials of operators and differential equations. Lyapunov stability. Limit cycles and chaos. Poincaré map. First integrals and integral manifolds. Structural stability and bifurcation. Characteristics of chaotic behaviour.
- Contents:
-
1. Basic theorem on the local existence and uniqueness of the solution
2. Continuous dependence and differentiability of the solution
3. Basics of the theory of autonomous systems
4. Analysis of solution of autonomous systems (special solutions, phase space)
5. Exponentials of operators and differential equations
6. Lyapunov stability
7. Limit cycles and chaos
8. Poincaré map
9. First integrals and integral manifolds
10. Structural stability and bifurcation
11. Characteristics of chaotic behaviour
- Seminar contents:
-
1. Basic theorem on the local existence and uniqueness of the solution
2. Continuous dependence and differentiability of the solution
3. Basics of the theory of autonomous systems
4. Analysis of solution of autonomous systems (special solutions, phase space)
5. Exponentials of operators and differential equations
6. Lyapunov stability
7. Limit cycles and chaos
8. Poincaré map
9. First integrals and integral manifolds
10. Structural stability and bifurcation
11. Characteristics of chaotic behaviour
- Recommended literature:
-
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, Springer-Verlag, Berlin 1990
[3] J. Guckenheimer and P.J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, Berlin 1983
Recommended references:
[3] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, Berlin 2003
- Keywords:
- Ordinary differential equations, qualitative theory, parmeter dependence, autonomous systems, limit cycles, Poincaré map
Abbreviations used:
Semester:
- W ... winter semester (usually October - February)
- S ... spring semester (usually March - June)
- W,S ... both semesters
Mode of completion of the course:
- A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
- GA ... Graded Assessment (a grade is awarded for this course)
- EX ... Examination (a grade is awarded for this course)
- A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)
Weekly load (hours per week):
- P ... lecture
- C ... seminar
- L ... laboratory
- R ... proseminar
- S ... seminar