MATH588 ORDINARY DIFFERENTIAL EQUATIONS II
| Course Code: | 2360588 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. MARAT AKHMET |
| Offered Semester: | Fall Semesters. |
Course Objectives
Ordinary differential equations are crucial in science and engineering as they offer mathematical models for various real-world phenomena. This significance underscores the need to enhance an advanced study of differential equations. The course focuses on three essential topics within the modern theory of motion: dynamical systems, emphasizing periodic motions; integral manifolds, which include stable, unstable, and center manifolds; and recurrent motions, such as almost periodic and Poisson stable motions, as well as bifurcations.
Course Content
Nonlinear Periodic Systems: Limit Sets; Poincare-Bendixon Theorem. Linearization Near Periodic Orbits; Method of Small Parameters in Noncritical Case; Orbital stability. Bifurcation: Bifurcation of Fixed Points; The Saddle-Node Bifurcation; The Transcritical Bifurcation; The Pitchfork Bifurcation; Hopf Bifurcation; Branching of Periodic Solutions for Nonautonomous Systems. Boundary Value Problems: Linear Differential Operators; Boundary Conditions; Existence of Solutions of BVPs; Adjoint Problems; Eigenvalues and Eigenfunctions for Linear Differential Operators; Green's Function of a Linear Differential Operator.
Course Learning Outcomes
By the end of the course, the student will learn the following components of the theory:
- Autonomous differential equations as dynamical systems: continuous dependence on the initial state, continuation, and the group property. Limit sets.
- Autonomous systems with periodic motions: Poincaré-Bendixson Theorem; Linearization Near Periodic Orbits; Orbital stability.
- Integral manifolds as a method of dimension reduction. Theorems on the existence of stable and unstable manifolds. The center manifold theorem.
- Bifurcation as a change of topological structures: the Saddle-Node Bifurcation, the Transcritical Bifurcation, the Pitchfork Bifurcation, and the Hopf Bifurcation.
- Method of Small Parameters in Critical and Noncritical Cases.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution