| MATHEMATICS (TURKISH, PHD) | |||||
| PhD | TR-NQF-HE: Level 8 | QF-EHEA: Third Cycle | EQF-LLL: Level 8 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT6019 | Dynamic Systems | Fall | 3 | 0 | 3 | 8 |
| The course opens with the approval of the Department at the beginning of each semester |
| Language of instruction: | Tr |
| Type of course: | Departmental Elective |
| Course Level: | |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Prof. Dr. CANAN ÇELİK KARAASLANLI |
| Course Objectives: | The aim of this course is to convey the required concepts and skills to design, model, and simulate dynamic systems. |
|
The students who have succeeded in this course; The students who succeded in this course: will be able to: Develop dynamical systems to model problems from biology, physics, and other areas. Analyze dynamical systems; apply computer methods to solve and visualize more complex systems. Determine the long term behavior of dynamical systems. Set up, analyze, and interpret phase portraits of linear and non-linear systems of differential equations Use graphical and symbolic methods to represent and interpret chaotic dynamical systems. |
| Autonomous equations and systems ( Function spaces and orbits, critical points and linearization, Liouville theorem), Linear and non-linear systems and their critical points in two dimensions. Periodic solutions (Bendixon Condition, Poincare-Bendixon theorem). Stability (Stability of equilibrium solutions and periodic solutions). Linear equations ( Linear equations of constant and periodic coefficients). Bifurcation Theory (Center manifold, Normal forms and Local bifurcations) |
| Week | Subject | Related Preparation | |
| 1) | Higher dimensions: The Lorenz system and chaos | ||
| 1) | Introduction to modeling and simulation | ||
| 2) | Linear dynamic systems; discrete and continuous time. | ||
| 3) | Nonlinear systems: fixed points, stability and linearization. | ||
| 4) | Lyapunov functions | ||
| 5) | Periodicity and Chaos | ||
| 6) | The Poincare-Bendixon Theorem | ||
| 7) | Hopf bifurcation | ||
| 8) | Periodicity in discrete time and stability of periodic points. | ||
| 9) | Nonlinear Techniques : Hamiltonian systems | ||
| 10) | Closed orbits and limit sets | ||
| 11) | Dynamical systems from biology. | ||
| 12) | Applications in Mechanics, Conservative systems. | ||
| 14) | The Lorenz system | ||
| Course Notes: | 1- Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second Edition (Pure and Applied Mathematics) by Stephen Smale, Morris W. Hirsch and Robert L. Devaney (Nov 5, 2003) 2-Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity) by Steven H. Strogatz. 3-Differential Equations and Dynamical Systems (Second Edition) by Lawrence Perko, published by Springer (1996). |
| References: |
| Semester Requirements | Number of Activities | Level of Contribution |
| Attendance | % 0 | |
| Laboratory | % 0 | |
| Application | % 0 | |
| Field Work | % 0 | |
| Special Course Internship (Work Placement) | % 0 | |
| Quizzes | % 0 | |
| Homework Assignments | 3 | % 10 |
| Presentation | % 0 | |
| Project | % 0 | |
| Seminar | % 0 | |
| Midterms | 1 | % 40 |
| Preliminary Jury | % 0 | |
| Final | 1 | % 50 |
| Paper Submission | % 0 | |
| Jury | % 0 | |
| Bütünleme | % 0 | |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 50 | |
| PERCENTAGE OF FINAL WORK | % 50 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Laboratory | 0 | 0 | 0 |
| Application | 0 | 0 | 0 |
| Special Course Internship (Work Placement) | 0 | 0 | 0 |
| Field Work | 0 | 0 | 0 |
| Study Hours Out of Class | 3 | 20 | 60 |
| Presentations / Seminar | 0 | 0 | 0 |
| Project | 0 | 0 | 0 |
| Homework Assignments | 3 | 15 | 45 |
| Quizzes | 0 | 0 | 0 |
| Preliminary Jury | 0 | ||
| Midterms | 1 | 23 | 23 |
| Paper Submission | 0 | ||
| Jury | 0 | ||
| Final | 1 | 30 | 30 |
| Total Workload | 200 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution | |
| 1) | Ability to assimilate mathematic related concepts and associate these concepts with each other. | |
| 2) | Ability to gain qualifications based on basic mathematical skills, problem solving, reasoning, association and generalization. | |
| 3) | Be able to organize events, for the development of critical and creative thinking and problem solving skills, by using appropriate methods and techniques. | |
| 4) | Ability to make individual and team work on issues related to working and social life. | |
| 5) | Ability to transfer ideas and suggestions, related to topics about his/her field of interest, written and verball. | |
| 6) | Ability to use mathematical knowledge in technology. | |
| 7) | To apply mathematical principles to real world problems. | |
| 8) | Ability to use the approaches and knowledge of other disciplines in Mathematics. | |
| 9) | Be able to set up and develope a solution method for a problem in mathematics independently, be able to solve and evaluate the results and to apply them if necessary. | |
| 10) | To be able to link abstract thought that one has to concrete events and to transfer the solutions and examine and interpret the results scientifically by forming experiments and collecting data. | |
| 11) | To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively. | |
| 12) | To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself. |