Math 123: Ordinary Differential Equations
Instructor: Di Fang Please read the Syllabus carefully!
Lecture: Tuesdays and Thursdays, 2:00--3:29pm, 3111 Etcheverry ⇒ Berkeley time!
Office Hours: Tue 3:40-4:39pm and Wed 3:00-3:59pm, 843 Evans Hall ⇒ Not Berkeley time!
Textbook: The Qualitative Theory of Ordinary Differential Equations by Fred Brauer and John A. Nohel.
Supplementary materials: L. Perko, Differential equations and dynamical systems. (a number of topics will follow this book and be summarized in my notes. So one does not need to purchase this book.)
Email: difang@berkeley.edu ⇒ Please kindly include Math 123 in the subject line when emailing.
Course Outline:
- Teminology; Review: techniques and tricks; the need for theory
- Theory of ODEs: existence, uniqueness, continuation.
- Linear Systems (with an intro to phase space analysis)
- Stability theory and Lyapunov method
- Basic dynamical systems, flow, chaos
- Applications and additional topics as time permits, e.g. biological models (flocking of birds), control theory.
This is a proof-based class. We will go over a number of theorems on the fundamental theory of ODEs, and also learn a number of basic analysis techniques -- such as the perturbation method, the fixed point argument, the bootstrapping argument -- that could be useful for future study in analysis.
(Things are subject to change. Once it happens I will let you know.)
| Week | Tuesday Topics | Thursday Topics | HW (posted on bCourse) |
|---|---|---|---|
| 1 | Policy, Philosophy, Terminology (not all in book) Lecture Notes |
None! Happy labor day! |
|
| 2 | How to solve ODEs? A Review of some techniques from [Calc/54] -- Old and New Lecture Notes | Review of techniques continued, some basic theorem/proof of the structure of the solution space, the need for theory [BN 1.5] Lecture Notes |
HW1 (due Sept 10th) |
| 3 | (Picard's) Existence theorem & proof (under the Lipschitz condition) [BN 3.1] -- Picard iteration, Uniform Convergence | Finish proof & (Peano's) Existence theorem, generalization to higher Dimension [BN 3.2], Gronwall Ineq [BN 1.7] & Uniqueness Theorem [BN 3.3] |
HW2 (due Sept 19th) |
| 4 | Continuation of Solutions, Global existence, a priori estimate [BN 3.4] | Dependence on initial condition and parameters [BN 3.5], Baby sensitivity analysis of Uncertainty Quantification, How equilibrium depends on parameter -- an intro to bifurcation analysis Lecture Notes |
HW3 (due Sept 26th) Chaotic system (sensitivity of initial data) -> try it here! |
| 5 | Existence theorem of system of linear ODEs [BN 2.2], vector norm, matrix norm [BN 1.4/2.1], a review of vector space
Lecture Notes |
Solution space of linear homogeneous systems, fundamental matrix, Abel's formula [BN 2.3] (on textbook, lecture notes will not be posted) | HW4 - fixed typo (due Oct 3rd) |
| 6 | Non-homogeneous system, Variation of constant formula [BN 2.4] | Linear system with constant coefficients, matrix exponenetial [BN 2.5] | HW5 (due Oct 10th) |
| 7 | More on complex eigen-pairs (using geometric meaning of rotation matrices); More on Jordan Block partial Lecture Notes |
No Class due to Power Outrage | No HW! Midterm is coming!! |
| 8 | Midterm | Fundamental matrix with general case - Jordan Canonical form, generalized eigenvectors, Jordan chains [BN 2.6] & on Linear Algebra! | No HW! |
| 9 | Asymptotic Behavior of Solutions of Linear System with Constant Coefficients -- estimate for homogeneuous & non-homogeneous cases [BN 2.7] Lecture Notes | Q&A on Midterm | HW6 (due Oct 31st) Happy Halloween! |
| 10 | A taste of coercivity, 2D linear autonoumous system (an intro to Phase space analysis) [BN 2.8] | Linear system with periodic coefficients, Floquent Theorem, Floquet multiplier [BN 2.9] | HW7 (due Nov 7th) Happy Halloween! |
| 11 | Finish linear system with periodic coefficients (non-homo case), periodic solutions [BN 2.9], intro to Lyapunov stability [BN 4.1] | stability theorem for linear constant coefficient system, variable coefficients -- perturbation method [BN 4.2-4.3] | HW8 (due Nov 14th) |
| 12 | Stability of almost linear system - linearization + perturbation [BN 4.4], bootstrapping argument Lecture Notes | Topological conjugate, Hartman-Grobman theorem (Ref: L. Perko, Differential equations and dynamical systems Chapter 2.8) | HW9 (due Nov 21th) |
| 13 | Lyapunov direct method, theorem proof [BN 5.2, 5.3] | Lyapunov method continued, LaSalle Invariance Principle, Hamiltonian system | HW10 (due Dec 3rd) Last homework of this semester! |
| 14 | Biological applications: Flocking of the birds (Reference Papers: [Tadmor-Ha 2008], [Liu-Ha 2009]) | HAPPY THANKSGIVING! | |
| 15 | Flocking Estimate continued (quantatitive rate via Dissipative inequality and Lyapunov) | Review and Q&A (let me know of the topics that are most confusing, and would like to go over again) | |
True or False: I do not need to take notes in class, because Di will always post her notes.
False!! Di is posting the notes for the first three lectures for easier reference because it was a review and material is not in the book. So take notes after the first three lectures.
(By Di on Sept. 3)
Wecome to the class. Wish we have a pleasant and productive semester together!
Please email me about the time conflicts with the office hour by Monday (Sept. 1st), so that we could settle the final office hours.
(By Di on August. 30)