Math 6307   Ordinary differential Equations I.   Fall 2017

 Schedule: 1:30 - 2:45 pm  TR Classroom: Skiles 257 Office: Skiles  102A Phone: 404-894-4750 Email: zengch@math.gatech.edu Webpage: www.math.gatech.edu/~zengch Office hours: 1:30 - 3 pm M & 4:05 - 4:40 pm T

Syllabus

Instructor: Chongchun Zeng

Prerequisites: MATH 2552&4541, or permission of the instructor

Participation: Attendence in classes is mandatory. Students are expected to read the materials before and/or after the classes. It is suggested that you start to work on homework problems right after they are assigned.

Main reference book: Ordinary differential equations with applications, Carmen Chicone, 2nd edition, Springer. (Homework will mainly come from this book.)

Materials:  In this semester, we will develop the qualitative theory for systems of ordinary differential equations. Subjects will are rough limit sets, stability, Lyapunov functions, linear systems, Floquet theory, attractors, invariant manifolds, bifurcation theory, normal forms, periodic orbits. The topics are among

• General properties: existence, uniqueness, continuous dependence, stability, Lyapunov functions, limit sets, Poincare-Bendixson theorem

• Linear Systems: constant coefficient systems, periodic systems, nonhomogeneous systems, Fredholm alternative, Hamiltonian systems, mappings

• Local theory near equilibria: Hartman-Grobman theorem, stable and unstable manifolds, foliations, center manifolds, elementary bifurcations and the Hopf bifurcation

• Periodic orbits: Poincare map, persistence, Hamiltonian systems.

• 1 take-home final exam 40%: `open-textbook' and `open-notes', must be entirely independent work.
• 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 are:

9/7(Thu.), 9/28(Thu.), 10/19(Thu.), 11/9(Thu.)

Homework assignments:

• Homework Assignment #1 (due Thu. 9/7):
• Assigned on Tue. 8/22:   Page 4.,   1.6 (aacde);  Page  5.,  1.10
• Assigned on Thu. 8/24: Attachment
• Assigned on Tue. 8/31:   Page 5.,  1.8;  Page 11.,  1.13
• Homework Assignment #2 (due Thu. 9/28):
• Assigned on Tue. 9/5:  Attachment
• Assigned on Thu. 9/7:  None.
• Assigned on Thu. 9/14:  Attachment.
• Assigned on Tue. 9/19:  Page 15,  1.20;  Page 16,  1.22, 1.25, 1.26;  Page 93,  1.169, 1.170
• Assigned on Thu. 9/21:  None.
• Homework Assignment #3 (due Thu. 10/19):
• Assigned on Tue. 9/26:  Page 32,  1.61(2, 5), 1.62(4, 6)
• Assigned on Thu. 9/28:  Page 99,  1.182;  Page 100,  1.184
• Assigned on Tue. 10/3:  Page 100,  1.186
• Assigned on Thu. 10/5:  Page 158,  2.25(1,4), 2.30(optional)
• Sketch of the solutions to some graded problems
• Homework Assignment #4 (due Thu. 11/9):
• Assigned on Thu. 10/12:  Page 153,  2.19(optional);  Page 158,  2.26, 2.27
• Assigned on Tue. 10/17:  Page 153,  2.20;  Page 159,  2.29, 2.30(optional);  Page 163,  2.34(optional);  Page 169,  2.45
• Assigned on Thu. 10/19:  Page 163,  2.34(a) (optional);   Page 167,  2.43
• Assigned on Tue. 10/24:  Page 193,  2.85, 2.86 (optional), 2.87 (optional)
• Assigned on Thu. 10/26: none.
• Assigned on Tue. 10/31:  Page 197,  2.93(optional)
• Sketch of the solutions to some graded problems
• Homework recommedned but not collected and not counted towards the final score:
• Assigned on Tue. 11/7:  Page 358.,  4.10
• Assigned on Thu. 11/16:  Page 347.,  4.4