常微分方程(一)

微分方程為數學及應用數學領域的基本課程,亦為應用科學及工程領域之基本課程。理工科系學生在大學課程裡都學過微分方程。此一研究所的常微分方程課程著重於引入更進一步的數學分析及幾何的概念及理論,來對常微分方程解的行為及結構,做更深入的探討。在很多應用領域,以微分方程呈現的數學模型多為非線性的微分方程,非線性微分方程基本上沒辦法解出明確解,需依賴電腦計算來解出數值解;數值解可以圖形來顯示、刻劃其性質,但是若超過三維空間(3個未知變量),解的軌道無法全然視覺化,只能畫出分量圖或投影圖,因此更需要數學理論及方法來解讀數值模擬,來探討、描述解的性質。此外,在三維空間中常微分方程解的軌道已可能有如混沌等複雜行為,不易刻劃。這門課亦會使用 MATLAB 計算常微分方程的解並解讀計算結果。在應用方面,我們會介紹生物、電路、類神經網路等數學模型。在Deep Neural Network的 continuous-depth models亦有用到ODE的架構。 ODE is a standard course in mathematics or applied mathematics graduate program. It is also a basic course in various applied science and engineering disciplines. In general, nonlinear differential equations can not be solved into exact forms. However, it is still feasible to understand at least a certain part of the dynamics or draw the phase portraits for nonlinear systems, through applying various mathematical ideas, concepts, and numerical simulations. The goal of this course is to learn the fundamental theories of ODE. They will be also useful in combining with numerical simulations when considering mathematical models in sciences and applied sciences. In tis course, we shall also introduce mathematical models on some biological problems, circuits, and neural networks. 1. Review undergraduate ODE 2. Autonomous linear systems 3. Non-autonomous linear systems and variations of constant formula 4. Fundamental theorems for ODE 5. Limit sets and attractors 6. Planar Systems, Poincare-Bendixson theory 7. Behaviors near equilibrium and Stability of equilibrium 8. Hamiltonian systems and gradient systems 9. Hartman-Grobman Theorem 10. Stable-Unstable manifolds

微積分 線性代數

使用E3教學平台:補充說明、補充教材、討論與回應將置於E3

模式1: 作業 35% 期中考 30% 期末考 35% 模式2: 作業 35% 上台發表、參與討論 30% 期末報告 35%

Lecture Notes, by Chih-Wen Shih