Differential Equations and Chaos
Code  Completion  Credits  Range 

01DRCH  Z  2  0+2 
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

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.
 Requirements:
 Syllabus of lectures:

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
 Syllabus of tutorials:

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
 Study Objective:

Knowledge:
Geometric theory of ordinary differential equations, Lyapunov stability, limit cycles, Poincaré map, structural stability, bifurcation, attractor.
Acquired skills:
Geometric analysis of problems for nonlinear ordinary differential equation. Analysis of asymptotic behaviour of solution. Properties of limit sets.
 Study materials:

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, SpringerVerlag, Berlin 1990
[3] J. Guckenheimer and P.J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, SpringerVerlag, Berlin 1983
Recommended references:
[3] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, SpringerVerlag, Berlin 2003
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans: