課程說明
Course Description
We will study differential equations from the perspective of dynamical systems, focusing on qualitative analysis of phase portraits. Bifurcation theory aims to analyze how topological changes in phase portraits occur as parameters vary. This theory has various applications, including but not limited to engineering (e.g.,beam buckling), biology (e.g., disease spread), chemistry (e.g., oscillatory reactions), physics (e.g., phase transitions), and climatology (e.g., global warming).
課程目標
Course Objectives
| 1. | Become proficient in studying differential equations from the perspective of dynamical systems. |
| 2. | Master qualitative analysis and rigorously apply concepts in bifurcation theory, such as center manifolds, |
| normal forms, and various reduction techniques. | |
| 3. | Explore the historical development and significant applications of dynamical systems |
| and differential equations. |
參考用書
References (not textbooks, sorted alphabetically)
教學進度
Course schedule
| Week | Date | Content |
| 1 | 02/19 | Flows and differential equations; |
| Transformation of vector fields | ||
| 2 | 02/26 | Linear autonomous ODEs; |
| Example: Poincaré diagram; Flow-box theorem | ||
| 3 | 03/05 | Stable manifolds and unstable manifolds |
| 4 | 03/12 | Center manifolds (applications) |
| 5 | 03/19 | Center manifolds (proof) |
| 6 | 03/26 | Normal forms |
| 7 | 04/02 | # School Activity Day (No Make-Up Classes) |
| 8 | 04/09 | Equivariant normal form theorem; |
| Procedure of local bifurcation analysis | ||
| 9 | 04/16 | Example: Planar Hopf bifurcation; |
| Example: Takens-Bogdanov bifurcation | ||
| 10 | 04/23 | Lyapunov-Schmidt reduction; |
| Stationary bifurcation | ||
| 11 | 04/30 | Example: Euler's rod; |
| Equivariant Lyapunov-Schmidt reduction | ||
| 12 | 05/07 | Stationary symmetry breaking bifurcation; |
| Example: Semilinear elliptic equations | ||
| 13 | 05/14 | Hopf bifurcation |
| 14 | 05/21 | Reversible Hopf bifurcation; |
| Equivariant Hopf bifurcation | ||
| 15 | 05/28 | Introduction of bifurcation without parameters; |
| Example: Transcritical bifurcation without parameters | ||
| 16 | 06/04 | 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. |
