Course detail

Simulation of non-linear dynamic systems

FAST-DU54Acad. year: 2020/2021

Definition of dynamic system, phase space, status variables, configuration space, solution of dynamic system, trajectories, fixed points, phase portrait, autonomous systems, operator of flow.
Planar linear systems analysis, canonical systems, criteria of classification, reduction of configuration space, qualitative equivalence, topological classification of planar systems, 3D systems, diffeomorfisms, first return map of Poincare, eigenspaces, hyperbolic and non-easy points.
Non-linear systems, local linearization, asymptotic and neutral stability, hyperbolic orbits, tangent spaces, limit set, limit cycle, attractor. Attracting set, Ljapunov exponents, homoclinic and heteroclinic structures, structural stability and bifucations, bifurcation diagrams, period doubling, chaotic behaviour.
Fractal dimension of strange attractors and boundaries of basins of attraction, symbolic dynamic, non-linear time series analysis, physical context.

Language of instruction

Czech

Guarantor

doc. RNDr. Jiří Macur, CSc.

Department

Institute of Computer Aided Engineering and Computer Science (AIU)

Learning outcomes of the course unit

Not applicable.

Prerequisites

theoretical mechanics, kinetic theory, basics of set theory, linear algebra, basic numeric methods, programming in OOP language

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Basic terminology - definition of continuous dynamic system, phase space, status variables, configuration space, solution of dynamic system, trajectories.
2. Fixed points, phase portrait, autonomous systems, using of operator of flow.
3. Planar linear systems analysis, canonical systems, criteria of classification, reduction of configuration space, qualitative equivalence, topological classification of planar systems
4. 3D systems, diffeomorfisms, first return map of Poincare, eigenspaces, hyperbolic and non-easy points.
5. Non-linear systems, local linearization, asymptotic and neutral stability, hyperbolic orbits, tangent spaces, limit set, limit cycle, attractor.
6. Attracting sets and basins of attraction, period doubling, Ljapunov exponents, chaotic behaviour.
7. Conservative and dissipative systems and their differences from viewpoint of evolution.
8. Homoclinic and heteroclinic structures, structural stability and bifucations, bifurcation diagrams.
9. Fractal dimension of strange attractors and boundaries of basins of attraction.
10. Symbolic dynamic, non-linear time series analysis.
11. Basics of numerical methods and algorithms for observing dynamic system behaviuor.
12-13. Simulation of physical dynamic systems, investigation of their non-linear phenomena.

Work placements

Not applicable.

Aims

Introduction to Dynamic Systems Theory, programming skills for simulation of dynamic systems evolution and analysis of their non-linear phenomena

Specification of controlled education, way of implementation and compensation for absences

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Nicolis, G.: Introduction to non-linear science. Cambridge Univ. Press 1995

Recommended reading

Macur, J.: Úvod do teorie dynamických systémů a jejich simulace. PC-DIR, Brno 1996
Macur, J.: Simulace dynamických systémů v jazyce Java. FAST VUT elektronický učební text 2001

Type of course unit

 

Lecture

39 hours, optionally

Teacher / Lecturer

doc. RNDr. Jiří Macur, CSc.

Syllabus

1. Basic terminology - definition of continuous dynamic system, phase space, status variables, configuration space, solution of dynamic system, trajectories. 2. Fixed points, phase portrait, autonomous systems, using of operator of flow. 3. Planar linear systems analysis, canonical systems, criteria of classification, reduction of configuration space, qualitative equivalence, topological classification of planar systems 4. 3D systems, diffeomorfisms, first return map of Poincare, eigenspaces, hyperbolic and non-easy points. 5. Non-linear systems, local linearization, asymptotic and neutral stability, hyperbolic orbits, tangent spaces, limit set, limit cycle, attractor. 6. Attracting sets and basins of attraction, period doubling, Ljapunov exponents, chaotic behaviour. 7. Conservative and dissipative systems and their differences from viewpoint of evolution. 8. Homoclinic and heteroclinic structures, structural stability and bifucations, bifurcation diagrams. 9. Fractal dimension of strange attractors and boundaries of basins of attraction. 10. Symbolic dynamic, non-linear time series analysis. 11. Basics of numerical methods and algorithms for observing dynamic system behaviuor. 12-13. Simulation of physical dynamic systems, investigation of their non-linear phenomena.