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Chaos and Non-Linear Phenomena, 6 credits
Kaos och icke-linjära fenomen, 6 hp
TFYA36
Main field of study
Mathematics Applied Mathematics Applied Physics PhysicsCourse level
Second cycleCourse type
Programme courseExaminer
Magnus JohanssonDirector of studies or equivalent
Magnus JohanssonEducation components
Preliminary scheduled hours: 56 hRecommended self-study hours: 104 h
Available for exchange students
YesECV = Elective / Compulsory / Voluntary
Main field of study
Mathematics, Applied Mathematics, Applied Physics, PhysicsCourse level
Second cycleAdvancement level
A1XCourse offered for
- Applied Physics and Electrical Engineering, M Sc in Engineering
- Physics and Nanoscience, Master's programme
- Applied Physics and Electrical Engineering - International, M Sc in Engineering
Specific information
The course is offered every second year. It will not be available during 2017.
Entry requirements
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.
Prerequisites
Linear algebra, Modern Physics, Classical Mechanics and some knowledge of Fourier Analysis and Partial Differential Equations. Analytical Mechanics, Quantum Mechanics and Physics of Condensed Matter are recommended.Intended learning outcomes
The aim of the course is that the students should get an orientation in the special properties of non-linear systems of theoretical and practical interest, and also an orientation in some topical areas of research.
After finishing the course the student knows how to:
- explain basic concepts in non-linear physics such as chaos, fractal dimensions, integrability, solitons and quantum chaos, and exemplify important applications of these concepts in different scientific disciplines.
- model simple non-linear dynamical systems mathematically, and demonstrate some of their characteristic properties by analyzing bifurcations and calculating Lyapunov-exponents and fractal dimensions for their attractors.
- explain mechanisms for appearance of chaos in conservative systems and properties of the corresponding quantum mechanical systems, and solve problems by applying the theory for Hamiltonian systems on simple models.
- derive soliton solutions in special mathematical models for wave motion in non-linear media, and exemplify their physical significance.
Course content
Introduction. Experiments and simple models. Vibrations in mechanical systems and electrical circuits. Piecewise linear maps and deterministic chaos. Universal behavior of quadratic maps. Bifurcations. Poincaré maps. Lyapunov exponents. Fractal dimensions. Period-doubling. Homoclinic and heteroclinic orbits. The intermittency route to chaos. Strange attractors in dissipative dynamical systems. The transition from quasiperiodicity to chaos. Regular and irregular motion of conservative systems. Integrable and non-integrable systems. Solitons and breathers with applications. Chaos in quantum systems.
Teaching and working methods
The course consists of lectures and exercises solving problems.
Examination
| UPG2 | Oral Presentation of Solved Problems | 0.5 credits | U, G |
| UPG1 | Hand-in assignments | 5.5 credits | U, 3, 4, 5 |
Grades
Four-grade scale, LiU, U, 3, 4, 5Department
Institutionen för fysik, kemi och biologiDirector of Studies or equivalent
Magnus JohanssonExaminer
Magnus JohanssonCourse website and other links
http://www.ifm.liu.se/courses/tfya36/kaos.htmlEducation components
Preliminary scheduled hours: 56 hRecommended self-study hours: 104 h
Course literature
Additional literature
Compendia
- G. Ohlén/S. Åberg/P. Östborn, Chaos
Compendium, Lund
Other
Complementary material handed out during the course.
| Code | Name | Scope | Grading scale |
|---|---|---|---|
| UPG2 | Oral Presentation of Solved Problems | 0.5 credits | U, G |
| UPG1 | Hand-in assignments | 5.5 credits | U, 3, 4, 5 |
Regulations (apply to LiU in its entirety)
The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.
LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva.
Additional literature
Compendia
G. Ohlén/S. Åberg/P. Östborn, Chaos
Compendium, Lund
Other
Complementary material handed out during the course.
Note: The course matrix might contain more information in Swedish.
I = Introduce, U = Teach, A = Utilize
| I | U | A | Modules | Comment | ||
|---|---|---|---|---|---|---|
| 1. DISCIPLINARY KNOWLEDGE AND REASONING | ||||||
| 1.1 Knowledge of underlying mathematics and science (G1X level) |
X
|
X
|
X
|
UPG1
UPG2
|
||
| 1.2 Fundamental engineering knowledge (G1X level) |
X
|
|
|
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| 1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level) |
|
|
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| 1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level) |
|
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| 1.5 Insight into current research and development work |
|
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| 2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES | ||||||
| 2.1 Analytical reasoning and problem solving |
|
X
|
X
|
UPG1
UPG2
|
||
| 2.2 Experimentation, investigation, and knowledge discovery |
|
|
|
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| 2.3 System thinking |
|
|
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| 2.4 Attitudes, thought, and learning |
|
X
|
X
|
UPG1
UPG2
|
||
| 2.5 Ethics, equity, and other responsibilities |
|
|
|
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| 3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION | ||||||
| 3.1 Teamwork |
|
|
X
|
|||
| 3.2 Communications |
|
X
|
X
|
UPG1
UPG2
|
||
| 3.3 Communication in foreign languages |
|
|
X
|
UPG1
UPG2
|
||
| 4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT | ||||||
| 4.1 External, societal, and environmental context |
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| 4.2 Enterprise and business context |
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| 4.3 Conceiving, system engineering and management |
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| 4.4 Designing |
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| 4.5 Implementing |
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| 4.6 Operating |
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| 5. PLANNING, EXECUTION AND PRESENTATION OF RESEARCH DEVELOPMENT PROJECTS WITH RESPECT TO SCIENTIFIC AND SOCIETAL NEEDS AND REQUIREMENTS | ||||||
| 5.1 Societal conditions, including economic, social, and ecological aspects of sustainable development for knowledge development |
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| 5.2 Economic conditions for knowledge development |
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| 5.3 Identification of needs, structuring and planning of research or development projects |
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| 5.4 Execution of research or development projects |
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| 5.5 Presentation and evaluation of research or development projects |
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