國立中興大學教學大綱
課程名稱 (中) 高等常微分方程(一)(5117)
(Eng.) Advanced Ordinary Differential Equation (I)
開課單位 應數系
課程類別 選修 學分 3 授課教師 戴佳原
選課單位 應數系 / 碩士班 授課使用語言 英文 英文/EMI Y 開課學期 1112
課程簡述 Dynamical systems are concerned with anything that changes. They originated from celestial mechanics and serve as the basis for the deterministic description of the universe. In this course we will study ordinary differential equations from the perspective of dynamical systems, with special emphasis on stability analysis of equilibrium and periodic solutions, invariant manifolds, and bifurcation theory.

This course is delivered in the framework of English as a Medium of Instruction (EMI). Mandarin is used only on occasions to ensure understanding and during the Q&A sessions. Both homework assignments and the final exam are all in English. Please note that the course content focuses on the mathematical theory and does not involve numerical simulations.

Suggested (but not mandatory) prerequisites: basic concepts of ordinary differential equations, an eagerness for learning, and an open mind. Regarding the basic concepts of ordinary differential equations, I recommend you watch the following videos before taking the course:
1. https://youtu.be/n3J_D4XEW_Y (equivalent to the course content in the second week)
2. https://youtu.be/hVPt_4dJDXs (equivalent to the course content in the fourth week and the first half of the fifth week)
3. https://youtu.be/Jn_Hb851v4U (equivalent to the course content in the second half of the fifth week and the first half of the sixth week)
4. https://youtu.be/9QHlhKcak5I (equivalent to the course content in the third week)
先修課程名稱
課程含自主學習 N
課程與核心能力關聯配比(%) 課程目標之教學方法與評量方法
課程目標 核心能力 配比(%) 教學方法 評量方法
1. Be familiar with studying ordinary differential equations from the viewpoint of dynamical systems.
2. Understand stability analysis, invariant manifolds, and bifurcation theory.
3: Know current state in the developments of differential equations.
1.數學專業思維與邏輯推理知識
2.數學分析專業知識
50
50
習作
討論
實習
講授
作業
測驗
授課內容(單元名稱與內容、習作/每週授課、考試進度-共18週)
週次 授課內容
第1週 * Self-directed learning: Videos for English for Specific Academic Purposes (ESAP)
第2週 Introduction: Three basic approaches to Study ODEs;
Flows and differential equations
第3週 Linear autonomous ODEs;
Example: Planar linear vector fields;
* Guidance for ignite talks
第4週 First integrals;
Example: Pendulum equations
第5週 Picard-Lindelöf theorem;
Maximal solutions
第6週 Smooth dependence;
Local stability criteria for equilibria
第7週 Floquet theory;
Local stability criteria for periodic orbits
第8週 Holiday
第9週 Conjugation of flows: Grobman-Hartman theorem
第10週 Stable manifolds and unstable manifolds
第11週 Center manifolds
第12週 Center manifolds (continue);
Application: Singular perturbations
第13週 Ignite talks;
First-order averaging theory
第14週 First-order averaging theory (continue)
第15週 Lyapunov-Schmidt reduction;
Stationary (steady-state) bifurcation
第16週 Stationary (steady-state) bifurcation (continue);
Example: Euler rod
第17週 Final exam
第18週 Discussion on the final exam
學習評量方式
Homework 50 %,
Ignite talk 20 %,
Final exam 30 %
教科書＆參考書目(書名、作者、書局、代理商、說明)
Recommended references (not textbooks, sorted alphabetically)
1. V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1988
2. V.I. Arnold: Ordinary Differential Equations, Springer, 2001
3. S.-N. Chow and J.K. Hale: Methods of Bifurcation Theory, Springer, 1982
4. M. Golubitsky and D.G. Schaeffer: Singularities and Groups in Bifurcation Theory, Volume 1, Springer, 1985
5. M. Golubitsky and Ian Stewart: The Symmetry Perspective, Springer, 2002
6. S.-B. Hsu: Ordinary Differential Equations with Applications, World Scientific Pub. Co. Inc., 2022
7. E. Zeidler: Nonlinear Functional Analysis and its Applications, Volume 1: Fixed-Point Theorems, Springer, 1998
課程教材（教師個人網址請列在本校內之網址）
Lecture notes (You can download them from the course webpage in iLearning 3.0)
課程輔導時間
Recitation Sessions (led by Prof. DAI Jia-Yuan)
Time: Thursday Session 1　
Place: Information Science Building Room 512

Office Hours
Time: Tuesday from 13:10 to 14:10
Place: Information Science Building Room 512
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更新日期 西元年/月/日：2023/04/16 17:15:19 列印日期 西元年/月/日：2025 / 7 / 06
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