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 BomanEducation components
Preliminary scheduled hours: 56 hRecommended self-study hours: 104 h
Available for exchange students
YesMain field of study
Mathematics, Applied Mathematics, Applied Physics, PhysicsCourse level
Second cycleAdvancement level
A1XCourse offered for
- Master's Programme in Physics and Nanoscience
- Master's Programme in Materials Science and Nanotechnology
- Physics and Nanoscience, Bachelor's Programme
- Applied Physics and Electrical Engineering - International, M Sc in Engineering
- Applied Physics and Electrical Engineering, M Sc in Engineering
Specific information
The course is offered every second year, even years. The course is not offered 2020.
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 but not necessary.
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. The course runs over the full spring term.
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, 5Other information
About teaching and examination language
The teaching language is presented in the Overview tab for each course. The examination language relates to the teaching language as follows:
- If teaching language is Swedish, the course as a whole or in large parts, is taught in Swedish. Please note that although teaching language is Swedish, parts of the course could be given in English. Examination language is Swedish.
- If teaching language is Swedish/English, the course as a whole will be taught in English if students without prior knowledge of the Swedish language participate. Examination language is Swedish or English (depending on teaching language).
- If teaching language is English, the course as a whole is taught in English. Examination language is English.
Other
The course is conducted in a manner where both men's and women's experience and knowledge are made visible and developed.
The planning and implementation of a course should correspond to the course syllabus. The course evaluation should therefore be conducted with the course syllabus as a starting point.
Department
Institutionen för fysik, kemi och biologiDirector of Studies or equivalent
Magnus BomanExaminer
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
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 |
Course syllabus
A syllabus must be established for each course. The syllabus specifies the aim and contents of the course, and the prior knowledge that a student must have in order to be able to benefit from the course.
Timetabling
Courses are timetabled after a decision has been made for this course concerning its assignment to a timetable module.
Interrupting a course
The vice-chancellor’s decision concerning regulations for registration, deregistration and reporting results (Dnr LiU-2015-01241) states that interruptions in study are to be recorded in Ladok. Thus, all students who do not participate in a course for which they have registered must record the interruption, such that the registration on the course can be removed. Deregistration from a course is carried out using a web-based form: https://www.lith.liu.se/for-studenter/kurskomplettering?l=en.
Cancelled courses
Courses with few participants (fewer than 10) may be cancelled or organised in a manner that differs from that stated in the course syllabus. The Dean is to deliberate and decide whether a course is to be cancelled or changed from the course syllabus.
Guidelines relating to examinations and examiners
For details, see Guidelines for education and examination for first-cycle and second-cycle education at Linköping University, http://styrdokument.liu.se/Regelsamling/VisaBeslut/917592.
An examiner must be employed as a teacher at LiU according to the LiU Regulations for Appointments (https://styrdokument.liu.se/Regelsamling/VisaBeslut/622784). For courses in second-cycle, the following teachers can be appointed as examiner: Professor (including Adjunct and Visiting Professor), Associate Professor (including Adjunct), Senior Lecturer (including Adjunct and Visiting Senior Lecturer), Research Fellow, or Postdoc. For courses in first-cycle, Assistant Lecturer (including Adjunct and Visiting Assistant Lecturer) can also be appointed as examiner in addition to those listed for second-cycle courses. In exceptional cases, a Part-time Lecturer can also be appointed as an examiner at both first- and second cycle, see Delegation of authority for the Board of Faculty of Science and Engineering.
Forms of examination
Examination
Written and oral examinations are held at least three times a year: once immediately after the end of the course, once in August, and once (usually) in one of the re-examination periods. Examinations held at other times are to follow a decision of the board of studies.
Principles for examination scheduling for courses that follow the study periods:
- courses given in VT1 are examined for the first time in March, with re-examination in June and August
- courses given in VT2 are examined for the first time in May, with re-examination in August and October
- courses given in HT1 are examined for the first time in October, with re-examination in January and August
- courses given in HT2 are examined for the first time in January, with re-examination in March and in August.
The examination schedule is based on the structure of timetable modules, but there may be deviations from this, mainly in the case of courses that are studied and examined for several programmes and in lower grades (i.e. 1 and 2).
Examinations for courses that the board of studies has decided are to be held in alternate years are held three times during the school year in which the course is given according to the principles stated above.
Examinations for courses that are cancelled or rescheduled such that they are not given in one or several years are held three times during the year that immediately follows the course, with examination scheduling that corresponds to the scheduling that was in force before the course was cancelled or rescheduled.
When a course is given for the last time, the regular examination and two re-examinations will be offered. Thereafter, examinations are phased out by offering three examinations during the following academic year at the same times as the examinations in any substitute course. If there is no substitute course, three examinations will be offered during re-examination periods during the following academic year. Other examination times are decided by the board of studies. In all cases above, the examination is also offered one more time during the academic year after the following, unless the board of studies decides otherwise.
If a course is given during several periods of the year (for programmes, or on different occasions for different programmes) the board or boards of studies determine together the scheduling and frequency of re-examination occasions.
Registration for examination
In order to take an examination, a student must register in advance at the Student Portal during the registration period, which opens 30 days before the date of the examination and closes 10 days before it. Candidates are informed of the location of the examination by email, four days in advance. Students who have not registered for an examination run the risk of being refused admittance to the examination, if space is not available.
Symbols used in the examination registration system:
** denotes that the examination is being given for the penultimate time.
* denotes that the examination is being given for the last time.
Code of conduct for students during examinations
Details are given in a decision in the university’s rule book: http://styrdokument.liu.se/Regelsamling/VisaBeslut/622682.
Retakes for higher grade
Students at the Institute of Technology at LiU have the right to retake written examinations and computer-based examinations in an attempt to achieve a higher grade. This is valid for all examination components with code “TEN” and "DAT". The same right may not be exercised for other examination components, unless otherwise specified in the course syllabus.
A retake is not possible on courses that are included in an issued degree diploma.
Retakes of other forms of examination
Regulations concerning retakes of other forms of examination than written examinations and computer-based examinations are given in the LiU guidelines for examinations and examiners, http://styrdokument.liu.se/Regelsamling/VisaBeslut/917592.
Plagiarism
For examinations that involve the writing of reports, in cases in which it can be assumed that the student has had access to other sources (such as during project work, writing essays, etc.), the material submitted must be prepared in accordance with principles for acceptable practice when referring to sources (references or quotations for which the source is specified) when the text, images, ideas, data, etc. of other people are used. It is also to be made clear whether the author has reused his or her own text, images, ideas, data, etc. from previous examinations, such as degree projects, project reports, etc. (this is sometimes known as “self-plagiarism”).
A failure to specify such sources may be regarded as attempted deception during examination.
Attempts to cheat
In the event of a suspected attempt by a student to cheat during an examination, or when study performance is to be assessed as specified in Chapter 10 of the Higher Education Ordinance, the examiner is to report this to the disciplinary board of the university. Possible consequences for the student are suspension from study and a formal warning. More information is available at https://www.student.liu.se/studenttjanster/lagar-regler-rattigheter?l=en.
Grades
The grades that are preferably to be used are Fail (U), Pass (3), Pass not without distinction (4) and Pass with distinction (5).
- Grades U, 3, 4, 5 are to be awarded for courses that have written examinations.
- Grades Fail (U) and Pass (G) may be awarded for courses with a large degree of practical components such as laboratory work, project work and group work.
- Grades Fail (U) and Pass (G) are to be used for degree projects and other independent work.
Examination components
- Grades U, 3, 4, 5 are to be awarded for written examinations (TEN).
- Examination components for which the grades Fail (U) and Pass (G) may be awarded are laboratory work (LAB), project work (PRA), preparatory written examination (KTR), oral examination (MUN), computer-based examination (DAT), home assignment (HEM), and assignment (UPG).
- Students receive grades either Fail (U) or Pass (G) for other examination components in which the examination criteria are satisfied principally through active attendance such as other examination (ANN), tutorial group (BAS) or examination item (MOM).
- Grades Fail (U) and Pass (G) are to be used for the examination components Opposition (OPPO) and Attendance at thesis presentation (AUSK) (i.e. part of the degree project).
For mandatory components, the following applies: If special circumstances prevail, and if it is possible with consideration of the nature of the compulsory component, the examiner may decide to replace the compulsory component with another equivalent component. (In accordance with the LiU Guidelines for education and examination for first-cycle and second-cycle education at Linköping University, http://styrdokument.liu.se/Regelsamling/VisaBeslut/917592).
For written examinations, the following applies: If the LiU coordinator for students with disabilities has granted a student the right to an adapted examination for a written examination in an examination hall, the student has the right to it. If the coordinator has instead recommended for the student an adapted examination or alternative form of examination, the examiner may grant this if the examiner assesses that it is possible, based on consideration of the course objectives. (In accordance with the LiU Guidelines for education and examination for first-cycle and second-cycle education at Linköping University, http://styrdokument.liu.se/Regelsamling/VisaBeslut/917592).
The examination results for a student are reported at the relevant department.
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.
Compendia
Compendium, Lund
Other
Complementary material handed out during the course.
Note: The course matrix might contain more information in Swedish.
I | U | A | Modules | Comment | ||
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1. DISCIPLINARY KNOWLEDGE AND REASONING | ||||||
1.1 Knowledge of underlying mathematics and science (G1X level) |
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X
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Basic courses in mathematics and physics |
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1.2 Fundamental engineering knowledge (G1X level) |
X
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Engineering applications within, e.g., mechanics and electronics |
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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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X
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UPG1
UPG2
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1.5 Insight into current research and development work |
X
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2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES | ||||||
2.1 Analytical reasoning and problem solving |
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X
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X
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UPG1
UPG2
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Problem solving; U: nonlinear mathematical models; A: basic physics and mathematics |
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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 |
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X
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UPG1
UPG2
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Creative and critical thinking (homework problems + problem presentation) |
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2.5 Ethics, equity, and other responsibilities |
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3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION | ||||||
3.1 Teamwork |
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X
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Problem presentations are often done in groups |
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3.2 Communications |
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X
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UPG1
UPG2
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Written homework problems and oral presentation of problems at the exercises |
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3.3 Communication in foreign languages |
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X
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UPG1
UPG2
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The course is given in English. |
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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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