Differential Equations and Chaos

Course guarantor:

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:

Basic knowledge of Calculus and Linear Algebra.

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, 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

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: