MATH589 IMPULSIVE DIFFERENTIAL EQUATIONS
| Course Code: | 2360589 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (0.00 - 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
The notion that discontinuity is a property of motion, alongside continuity, is as old as the concept of motion itself. Many mathematicians today agree that both continuity and discontinuity should be taken into account to describe the real world more accurately.
This course is primarily an introduction to the theory of discontinuous dynamical systems. There are some similarities between this course and those on ordinary differential equations. Therefore, we will cover several standard topics, including the description of systems, definitions of solutions, local existence and uniqueness theorems, extension of solutions, and the dependence of solutions on parameters. We will also explore theorems related to periodic solutions and conduct stability analysis.
Course Content
General description of impulsive differential equations: Systems with fixed moments of impulses; systems with variable moments of impulses; discontinuous dynamical systems. Linear systems: General properties of solutions; periodic solutions; Floquet theory; adjoint systems. Stability: Stability criterion based on linearization of systems; direct Lyapunov method; B-equivalence; stability of systems with variable time of impulses. Quasilinear systems: Bounded solutions; periodic solutions; quasiperiodic and almost periodic solutions; integral manifolds. Discontinuous dynamical systems and applications.
Course Learning Outcomes
By the end of the course, the student will learn the following components of the theory:
- The concept of impulsive differential equations as hybrid systems with continuous and discontinuous compartments
- Systems with fixed moments of impulses: Existence and uniqueness of solutions; Continuation of solutions
- Conditions for periodic dynamics; Linear and quasilinear systems; Floquet theory; Stability analysis
- Linearization; Continuous and differentiable dependence on the initial data and parameters
- The method of small parameters
- Systems with non-fixed moments of impulses; The beating phenomenon; B-equivalence method; Periodic dynamics; Lyapunov stability
- Almost periodic impulsive systems and discontinuous solutions
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