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 | 12 |
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 |