Module also offered within study programmes:
General information:
Name:
Nonlinear Dynamics
Course of study:
2016/2017
Code:
JFI-3-404-s
Faculty of:
Physics and Applied Computer Science
Study level:
Third-cycle studies
Specialty:
-
Field of study:
Physics
Semester:
4
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Responsible teacher:
prof. dr hab. Kułakowski Krzysztof (kulakowski@fis.agh.edu.pl)
Academic teachers:
Module summary

The lecture is intended to familiarize students with qualitative methods of analysis of nonlinear dynamics. Most of these issues concern ordinary differential equations.

Description of learning outcomes for module
MLO code Student after module completion has the knowledge/ knows how to/is able to Connections with FLO Method of learning outcomes verification (form of completion)
Skills
M_U001 students knows numerical methods to detect chaotic effects FI3A_U03 Execution of a project
M_U002 students knows qualitative methods of handling of nonlinear differential equations FI3A_U01, FI3A_U02 Execution of exercises
Knowledge
M_W001 student understands limitations of classical mechanics due to the deterministic chaos FI3A_W01, FI3A_W02 Examination
M_W002 Student understand questions and problems formulated in terms on NL FI3A_W01, FI3A_W02 Examination,
Participation in a discussion
FLO matrix in relation to forms of classes
MLO code Student after module completion has the knowledge/ knows how to/is able to Form of classes
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Others
E-learning
Skills
M_U001 students knows numerical methods to detect chaotic effects - - - - - - - - - - -
M_U002 students knows qualitative methods of handling of nonlinear differential equations - - - - - - - - - - -
Knowledge
M_W001 student understands limitations of classical mechanics due to the deterministic chaos + - - - - - - - - - -
M_W002 Student understand questions and problems formulated in terms on NL + - - - - - - - - - -
Module content
Lectures:
Properties of dynamical systems described with differential or difference equations. Qualitative methods.

1. Elementary methods of analysis of 2-dimensional problems: • Stability of fixed points • Linearization, Jordan forms • Constant of motion, isoclines, phase portrait • Approximated calculation of trajectories near a fixed point
2. Selected qualitative methods: • Lyapunov function and her applications. Bounding function • Diagram Determinant-Trace • Invariant manifold • Types of fixed points • Poincare indices and their properties • Divergence test. Dulac criterion • Poincare-Bendixon theorem • Landau symbols • Resonances • Poincare theorem on linearization
3. Approximated analytical methods: • Perturbation calculus • Method of two time scales
4. Bifurcations in differential equations: • Saddle-node bifurcation • Transcritical bifurcation • Pitchfork bifurcation • Hopf bifurcation
5. Bifurcations in difference equations: • Stability of fixed points in difference equations • Bifurcations in difference equations • Period-doubling bifurcation • Logistic equation
6. Elements of symbolic dynamics: • Sharkovskii ordering • Superstable cycles • Word Lifting technique • Structural universality • Arnold tongues • Farey tree and devil staircases
7. Data analysis: • Fractal dimension • Lyapunov indices. Li-Yorke conjecture • Experiment Fermi-Pasta-Ulam • Invariant measure. Frobenius-Perron equation • Correlation function • Bernoulli shift • Mixing • Deterministic diffusion • R/S analysis. Hurst law • Multifractals

Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 98 h
Module ECTS credits 4 ECTS
Preparation for classes 42 h
Participation in lectures 28 h
Realization of independently performed tasks 28 h
Additional information
Method of calculating the final grade:

final exam (at least satisfactory, 3.0)

Prerequisites and additional requirements:

Prerequisites and additional requirements not specified

Recommended literature and teaching resources:

Basic literature: P. Glendinning, Stability, instability and chaos, Cambridge UP 1994

Scientific publications of module course instructors related to the topic of the module:

Additional scientific publications not specified

Additional information:

None