Math 6308 Ordinary differential Equations II. Spring 2018
Schedule: 1:30 - 2:45
Classroom: Skiles 255
Office: Skiles 102A
Office hours: 10:05 - 11am W & 12:30 - 1:25pm R
Prerequisites: MATH 2552&4541, or permission of the instructor
classes is mandatory. Students are expected to read the materials
after the classes. It is suggested that you start to work on
problems right after they are assigned.
Main reference book:
Ordinary differential equations with
applications, Carmen Chicone, 2nd edition, Springer. (Homework will mainly
In this semester, we will develop the qualitative
theory for systems of nonlinear ordinary differential equations. After
a brief review of the material covered in Math 6307, the main subjects
include invariant manifold theory,
bifurcation theory, normal forms, periodic orbits and homoclinic
orbits. The topics are among
- Invariant manifolds: existence, uniqueness, smoothness, and other properties of invariant manifolds and invariant foliations.
- Basic bifurcations: saddle-node, transcritical, pitch-fork, and Hopf bifurcations.
- Periodic orbits:
Poincare map, persistence, period-doubling, and periodic orbtis in Hamiltonian systems.
- Homoclinic orbits: Melnikov method, Smale-Birkhoff theorem, etc.
- 1 take-home final exam 40%: `open-textbook' and `open-notes', must be entirely
- 4 homework, 15% each: homework will be assigned at the end of
each lecture and then put on this webpage in the below. You may discuss
the homework problems with other students in this class, but you should write down the solutions and complete the homework independently.
On each collecting day, the homework is due in the classroom at the
beginning of the lecture. The corresponding assignment includes all the
problems assigned in previous weeks, but after the last assignment.
Some selected problems from each assignment will be graded and the
score will be given based on both the graded problems and the
completion of the whole assignment. The exact homework collecting dates
1/25(Thu.), 2/15(Thu.), 3/8(Thu.), 4/5(Thu.)
- Homework Assignment #1 (due Thu. 1/25):
- Assigned on Tue. 1/9: P159, 2.30; P163, 2.34; P193, 2.86, 2.87
- Assigned on Thu. 1/11: P228, 3.2; P233, 3.5, 3.7
- Homework Assignment #2 (due Thu. 2/15):
- Homework Assignment #3 (due Thu. 3/8)
- Assigned on Tue. 2/13: Page 347, 4.4, 4.5
- Assigned on Thu. 2/22: Page 558, 8.9
- Homework Assignment #4 (due Thu. 4/5)
- Assigned on Tue. 3/6: Page 564, 8.18, 8.19
- Assigned on Tue. 3/13: P373, 5.2
- Assigned on Tue. 3/27: P382, 5.9