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 | Bachelor’s Degree (First Cycle) |
| 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 | |
|---|---|
| Major Area Courses | X |
| 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 | Acquires skills to use the advanced theoretical and applied knowledge obtained at the mathematics bachelors program to do further academic and scientific research in both mathematics-based graduate programs and public or private sectors. | X | ||||
| 2 | Transplants and applies the theoretical and applicable knowledge gained in their field to the secondary education by using suitable tools and devices. | X | ||||
| 3 | Acquires the skill of choosing, using and improving problem solving techniques which are needed for modeling and solving current problems in mathematics or related fields by using the obtained knowledge and skills. | X | ||||
| 4 | Acquires analytical thinking and uses time effectively in the process of deduction. | X | ||||
| 5 | Acquires basic software knowledge necessary to work in the computer science related fields and together with the skills to use information technologies effectively. | X | ||||
| 6 | Obtains the ability to collect data, to analyze, interpret and use statistical methods necessary in decision making processes. | X | ||||
| 7 | Acquires the level of knowledge to be able to work in the mathematics and related fields and keeps professional knowledge and skills up-to-date with awareness in the importance of lifelong learning. | X | ||||
| 8 | Takes responsibility in mathematics related areas and has the ability to work affectively either individually or as a member of a team. | 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 the ability to communicate ideas with peers supported by qualitative and quantitative data. | X | ||||
| 11 | Has professional and 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 | ||