Math 555: Ordinary Differential Equations (Fall 2025)
Instructor: Di Fang
Please find the Course webpage on Canvas! This page is generated from Canvas and intended solely for maintaining a record of the teaching schedule and may not be consistently updated.
Please find the references (textbooks) for the course in the syllabus.
Math 555 is a graduate-level ODE course that is theoretical and proof-oriented. It is not about solving ODEs, but about developing a deep understanding of their behavior and properties--without finding explicit solutions.
(Students who wish to learn how to solve ODEs should take undergraduate courses such as Math 356 or Math 353 instead. For numerical approaches to ODEs, please refer to courses in numerical analysis.)
Prerequisites: Linear algebra (the equivalent of 221); real analysis (the equivalent of 531 or 431); basic differential equation solving from calculus; an introductory course on differential equations (the equivalent of 353 or 356) strongly encouraged by not strictly required.
The course will cover:
- Part 1 Theory of Linear systems (Fundamental Matrix, Asymptotic Behavior, Variation of Constant Theorem, time-ordered matrix exponential, phase portrait, etc)
- Part 2 Fundamental Theory (Well-posedness) of ODEs (local existence, Picard iteration, Banach fixed point theorem/contraction principle, Peano's local existence; uniqueness theorem, comparison principle; global existence, a priori estimate; sensitivity/dependence on initial data and parameters, bifurcation)
- Part 3 Stability Theory (Linear system with constant coefficients; Floquet theory; Perturbation method (for both linear systems and nonlinear systems) + Linearization, bootstrapping argument; Grobman-Hartman Theorem, stable/unstable manifold; Lyapunov functions, La Salle's Principle)
- Part 4 2D system (Poincare-Bendixson Theorem, omega-limit set, limit cycles; Hopf bifurcation; Index theory in the plane)
- Part 5 More on Dynamical Systems (Hamiltonian systems and integrability; stability of periodic orbits via Poincare map; Intro to Chaos -- Devaney's chaos and strange attractor)
Tentative Course Plan: (subject to change)
Some handwritten notes may be posted (not guaranteed to be typo-free); or References will be provided.
* Remember, we are on the same team to learn and succeed. If you have any questions, I encourage you to come talk to me during office hours.
| Week | Wednesday | Friday | HW |
| 1 |
ODEs: old and new (motivation, definitions, tricks), revisit of basic techniques Math555_lec1.pdf
Download Math555_lec1.pdf |
Linear homogeneous ODEs: Solution structure, fundamental matrix, Abel's formula. Math555_lec2.pdf Download Math555_lec2.pdf |
HW 1 (due: Sept 12) |
| 2 |
Linear nonhomogeneous ODEs, variation of constant formula; more on fundamental matrix for time-dependent $A(t)$; normed vector space |
Linear ODEs with constant coefficients, matrix norm, matrix exponential and properties |
|
| 3 |
More on fundamental matrix with constant coefficients; complex eigen-pairs: compute fundamental matrix without detailed calculation; Revisit of Jordan chains and generalized eigenvector, and their implications to fundamental matrix; start of asymptotic behavior |
Asymptotic Behavior (large-time behavior) of the solutions of the linear ODEs with constant coefficients |
|
| 4 |
Phase Portraits for 2D systems (including a revisit of 2D jordan chains, generalized eigenvector; fundamental matrices for 3 cases, phase portrait of center and spiral, saddle, nodes) and classification of equilibrium. |
The Need for Theory; (Picard's) Local Existence Theorem: Statement, and a start of the Proof. Math555_lec8.pdf Download Math555_lec8.pdf |
|
| 5 |
(Picard's) Local Existence Theorem: Proof cont'd; Peano's local existence theorem; Gronwall's Inequality, Uniqueness Theorem; Revisit Banach space from real analysis. |
Contraction Principle/Banach Fixed Point Theorem; Proof Picard's local existence + uniqueness via Contraction Principle; contraction principle v.s. compactness argument; |
|
| 6 |
Continuation/Extensibility of Solutions, Global existence, a priori estimate |
Midterm (in class) |
HW5 (due Oct 10) |
| 7 |
Intro to Lyapunov stability, Stability of linear ODE with constant coefficients. Stability Method 1: Perturbation Method -- Case 1: for some time-dependent coefficients |
Stability Method 1: Perturbation Method cont'd -- Case 2: Stability of autonomous systems - (linearization + Perturbation). Bootstrapping argument Math555_lec13.pdf Download Math555_lec13.pdf |
HW6 (due Oct 17) |
| 8 |
Bootstrapping cont'd -- Comparison Principle (super/subsolutions); Dependence on initial condition and parameters; Baby sensitivity analysis of Uncertainty Quantification; How equilibrium depends on parameter -- an intro to bifurcation |
Linear ODEs with periodic coefficients, Floquet theory, periodic solutions Math555_lec15.pdf Download Math555_lec15.pdf |
HW7 (due Oct 24) |
| 9 |
Finish nonhomogeneous periodic systems (theorem on its periodic solutions); Topological conjugate, Hartman-Grobman theorem, hyperbolic fixed point, flow of autonomous systems |
orbits, Stable/unstable manifold theorem; Stability Method 2: Lyapunov Functions, Understanding and Examples Math555_lec17.pdf Download Math555_lec17.pdf |
HW8 (due Oct 31) |
| 10 |
Proofs of Lyapunov Theorems (stable case and asymptotically stable case); Twisted norm (a taste of coercivity vs hypocoercivity) |
LaSalle Invariance Principle; Newtonian system; Periodic orbits, Poincare-Bendixon theorem, trapping region and Van der Pol oscillator - part 1; negative Poincare criterion Math555_lec19.pdf Download Math555_lec19.pdf |
|
| 11 |
Van der Pol oscillator - cont'd; Limit cycle, $\omega$ (and $\alpha$) -limit set, the generalized Poincare-Bendixon theorem |
Index theory in the plane; winding number and index for a jordan curve and a fixed point; properties. Mentioning of the Hilbert's 16th problem; Hamiltonian system. (this lecture follows the book Differential Equations and Dynamical Systems by Perko) Math555_lec21.pdf Download Math555_lec21.pdf |
HW10 (due Nov 14) |
| 12 |
More on Hamiltonian systems; Phase portrait; A theorem of the fixed points of Newtonian systems; symplectic gradient, first integrals. |
Action-angle variables, canonical transformation, integrable v.s. non-integrable, a taste of KAM theory. Poincare section. Use Poincare section and Poincare map to study the stability of periodic orbits | No HW |
| 13 |
Hopf bifurcation; Intro to chaos -- Devaney's (topological) chaos, Lyapunov exponent, period doubling for logistic map, Sharkovski theorem and Li Yorke theorem (no proof), strange attractor. |
Final |