課程說明
Course Description
Delay differential equations are a type of differential equation where the rate of change of a system at a particular time depends not only on its current state but also on its state at previous times. This time delay can represent time lags in physical, biological, or other dynamical systems where the effect of an action is not instantaneous. We will study delay differential equations from the perspective of infinite-dimensional dynamical systems, exploring the impact of time delays on nonlinear phenomena. Specifically, we will first cover the existence, uniqueness, and stability of solutions, followed by important applications such as time-delayed feedback control and global bifurcation.
課程目標
Course Objectives
| 1. | Become familiar with studying delay differential equations from the perspective of dynamical systems. |
| 2. | Master the theories of existence, uniqueness, and stability of solutions. |
| 3. | Introduce important applications of delay differential equations, |
| such as time-delayed feedback control and global bifurcation. | |
參考用書
References (not textbooks, sorted alphabetically)
教學進度
Course schedule
| Week | Date | Content |
| 1 | 09/05 | Introduction to DDEs |
| 2 | 09/12 | Are DDEs finite- or infinite-dimensional? Viewpoint of dynamical systems |
| 3 | 09/19 | Existence, uniqueness, and continuation; Eventually compact semiflows |
| 4 | 09/26 | (Interlude) Local stability of ODE equilibria; Linear autonomous FDEs |
| 5 | 10/03 | # Typhoon Holiday |
| 6 | 10/10 | # Holiday |
| 7 | 10/17 | Characteristic equations; Infinitely many eigenvalues of linear discrete DDEs |
| 8 | 10/24 | Stability region; Spectral effect of small delays |
| 9 | 10/31 | # Typhoon Holiday |
| 10 | 11/07 | Spectral effect of large delays; Example: Neutral differential difference equations |
| 11 | 11/14 | Variation-of-constant formula for linear discrete DDEs |
| 12 | 11/21 | (Interlude) Local stability of ODE periodic orbits |
| 13 | 11/28 | Pyragas feedback control stabilization; Example: Subcritical Stuart-Landau oscillators; |
| Limitation of Pyragas control | ||
| 14 | 12/05 | DDEs of negative feedback type; Global attractors; Zero number |
| 15 | 12/12 | Morse Decomposition; Limit sets and Poincaré-Bendixson theorem; |
| Geometry of the set of periodic solutions Talk by Dr. Alejandro López-Nieto followed by a joint discussion | ||
| 16 | 12/19 | Final Exam |
評分標準
Grading Criteria
| ♠ | Homework 70 %, final exam 30 % |
注意事項
Important Notice
| ♠ | Plagiarism in assignments and cheating in exams are strictly prohibited in this course. |
| If substantial evidence confirms plagiarism in assignments or cheating in exams,the semester grade will be recorded as zero. |
生成式人工智慧倫理聲明
Generative AI Ethics Statement
| ♠ | This course allows students to use generative AI in their learning process, but be aware that |
| generative AI still contains many errors and may affect the understanding of fundamental core knowledge. When using generative AI, students must adhere to the ethical guidelines provided in the course. |
