Chaotic Systems and Their Analysis

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The course is devoted to analysis of chaotic systems by means of analytical and numerical methods.

Requirements:

Basic course of Calculus a Linear Algebra (in the extent of the courses 01MA1, 01MAB234, 01LA1, 01LAB2 held at the FNSPE CTU in Prague).

Syllabus of lectures:

Dynamics of differential equations: special functions, solving in quadrature, dynamics in phase space, analysis of linear stability, fixed points, limit cycles, non-autonomous systems. Hamilton dynamics: Lagrange function and properties, Hamilton formulation of dynamics, Hamilton equations, Poisson brackets, canonical transformations, optimal transformations, action-angle variable, integrable hamiltonians, examples. Perturbation theory: elementary perturbation theory, regular perturbation series, theory of canonical perturbations, KAM theorem, super-convergent perturbation theory, invariants of KAM theorem. Chaos in Hamiltonian systems: surface of section, Henon-Heiles hamiltonian, Toda lattice, symplectic mapping, area-preserving mappings, Poincare-Birkhoff theorem on fixed point, homoclinic a heteroclinic points, criteria for local chaos, Lyapunov exponents, criteria for widespread chaos, ergodicity, mixing, baker?s transformation, Bernoulli systems. Numerical simulations of chaotic systems: implicit numerical schemes, schemes of symplectic methods.

Syllabus of tutorials:

1. Analysis of differential equations. 2. Hamilton formulation of mechanics. 3. Perturbation theory. 4. Chaos in Hamiltonian systems. 5. Numerical simulations of chaotic systems.

Study Objective:

Knowledge: Analysis of linear stability for chosen systems, application of perturbation theory, features of widespread chaos, numerical analysis of chaotic systems. Skills: Individual analysis of linear stability of chosen system.

Study materials:

Key references:

[1] M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction, Wiley-Interscience, 1989,

[2] F. Haake, Quantum Signatures of Chaos, Springer 2000

Recommended references:

[3] H.J. Korsch, H.J. Jodl, Chaos, Columbia University, Springer-Verlag, Berlin, 1999

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: