ECTS - Dynamical Systems and Chaos
Dynamical Systems and Chaos (MATH467) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Dynamical Systems and Chaos | MATH467 | Area Elective | 4 | 0 | 0 | 4 | 6 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Natural & Applied Sciences Master's Degree |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Question and Answer. |
| Course Lecturer(s) |
|
| Course Objectives | The mathematical formulation of numerous physical problems results in differential equations which are actually nonlinear. This course is about dynamical aspects of nonlinear ordinary differential equations. It treates chiefly autonomous systems, emphasizing qualitative behavior of solution curves, and gives an introduction to the phase portrait analysis of such systems. |
| Course Learning Outcomes |
The students who succeeded in this course;
|
| Course Content | One-dimensional dynamic systems, stability of equilibria, bifurcation, linear systems and their stability, two-dimensional dynamic systems, Liapunov?s direct method and method of linearization, 3-dimensional dynamic systems. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | Linear Systems: Uncoupled Linear Systems, Diagonalization | pp. 1-6 |
| 2 | Exponentials of Operators, The Fundamental Theorem for Linear Systems, Linear Systems in R^2 | pp. 6-20 |
| 3 | Complex Eigenvalues, Multiple Eigenvalues | pp. 20-32 |
| 4 | Jordan Forms, Stability Theory, Nonhomogeneous Linear Systems | pp. 32-64 |
| 5 | Nonlinear Systems: Some Preliminary Concepts and Definitions, The Fundamental Existence-Uniqueness Theorem, Dependence on Initial Conditions and Parameters | pp. 65-79 |
| 6 | The Maximal Interval of Existence, The Flow Defined by a Differential Equation, Linearization | pp. 79-105 |
| 7 | Midterm | |
| 8 | The Stable Manifold Theorem, Stability and Liapunov Functions | pp. 105-119 and pp. 129-136 |
| 9 | Saddles, Nodes, Foci and Centers, Nonhyperbolic Critical Points in R^2, Center Manifold Theory | pp. 136-163 |
| 10 | Nonlinear Systems: Global theory, dynamical systems and global eExistence theorems, limit sets and attractors | pp. 181-202 |
| 11 | Periodic Orbits, Limit Cycles, The Stable Manifold Theorem for Periodic Orbits | pp. 202-211 and pp. 220-234 |
| 12 | Hamiltonian Systems, The Poincare-Bendixson Theory in R^2, Bendixson's Criteria | pp. 234-252 and pp. 264-267 |
| 13 | Nonlinear Systems: Bifurcation Theory, Structural Stability | pp. 315-334 |
| 14 | Bifurcations at Nonhyperbolic Equilibrium Points | pp. 334-343 |
| 15 | Review | |
| 16 | Final Exam |
Sources
| Course Book | 1. L. Perko, Differential Equations and Dynamical Systems: 3rd Edition, Springer, New York, 2000. |
|---|---|
| Other Sources | 2. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems: 2nd Edition, Springer, New York, 1996. |
| 3. M.W. Hirsch, S. Smale and R.L. Devaney, Differential Equations, Dynamical Systems and, An Introduction to Chaos: 2nd Edition, Academic Press, San Diego, 2004. | |
| 4. W. Kelley and A.Peterson, The Theory of Differential Equations: Classical and Qualitative, Pearson Education, New Jersey, 2004. | |
| 5. S.L.Ross, Differential Equations, 3rd edition, Wiley, New York, 1984 |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 2 | 20 |
| Presentation | - | - |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 40 |
| Final Exam/Final Jury | 1 | 40 |
| Toplam | 4 | 100 |
| Percentage of Semester Work | 60 |
|---|---|
| Percentage of Final Work | 40 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area. | X | ||||
| 2 | Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research. | X | ||||
| 3 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research. | X | ||||
| 4 | Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary. | X | ||||
| 5 | Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study. | X | ||||
| 6 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework. | X | ||||
| 7 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility. | X | ||||
| 8 | Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge. | X | ||||
| 11 | Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | |||
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 16 | 4 | 64 |
| Presentation/Seminar Prepration | |||
| Project | |||
| Report | |||
| Homework Assignments | 2 | 8 | 16 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 16 | 16 |
| Prepration of Final Exams/Final Jury | |||
| Total Workload | 96 | ||