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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025

Chaotic Systems and Their Analysis

The course is not on the list Without time-table
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, 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:
Data valid to 2024-05-23
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