Chaotic Systems and Their Analysis
Code  Completion  Credits  Range  Language 

01CHAOS  ZK  2  2+0  Czech 
 Garant předmětu:
 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, nonautonomous systems. Hamilton dynamics: Lagrange function and properties, Hamilton formulation of dynamics, Hamilton equations, Poisson brackets, canonical transformations, optimal transformations, actionangle variable, integrable hamiltonians, examples. Perturbation theory: elementary perturbation theory, regular perturbation series, theory of canonical perturbations, KAM theorem, superconvergent perturbation theory, invariants of KAM theorem. Chaos in Hamiltonian systems: surface of section, HenonHeiles hamiltonian, Toda lattice, symplectic mapping, areapreserving mappings, PoincareBirkhoff 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, WileyInterscience, 1989,
[2] F. Haake, Quantum Signatures of Chaos, Springer 2000
Recommended references:
[3] H.J. Korsch, H.J. Jodl, Chaos, Columbia University, SpringerVerlag, Berlin, 1999
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans: