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 Spring |
3 | 0 | 3 | 8 |
This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester. |
Language of instruction: | Turkish |
Type of course: | Departmental Elective |
Course Level: | |
Mode of Delivery: | Face to face |
Course Coordinator : | Prof. Dr. CANAN ÇELİK KARAASLANLI |
Recommended Optional Program Components: | Matlab |
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 / Textbooks: | 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 |
Homework Assignments | 3 | % 10 |
Midterms | 1 | % 40 |
Final | 1 | % 50 |
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 |
Study Hours Out of Class | 3 | 20 | 60 |
Homework Assignments | 3 | 15 | 45 |
Midterms | 1 | 23 | 23 |
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. | 5 |
2) | Ability to gain qualifications based on basic mathematical skills, problem solving, reasoning, association and generalization. | 5 |
3) | Be able to organize events, for the development of critical and creative thinking and problem solving skills, by using appropriate methods and techniques. | 5 |
4) | Ability to make individual and team work on issues related to working and social life. | 5 |
5) | Ability to transfer ideas and suggestions, related to topics about his/her field of interest, written and verball. | 4 |
6) | Ability to use mathematical knowledge in technology. | 4 |
7) | To apply mathematical principles to real world problems. | 4 |
8) | Ability to use the approaches and knowledge of other disciplines in Mathematics. | 4 |
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. | 5 |
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. | 5 |
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. | 5 |
12) | To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself. | 4 |