Math 489. Differential Equations II (formerly Math 385)
Syllabus for Instructors
Text: Arrowsmith and Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behavior, 1st Edition, Chapman and Hall/CRC, 1996.
The course should cover Chapters 1, 2, 3 and 5, and some parts of 4 and 6. The goal is to bring students to a point where they can learn more advanced topics on their own. So rather than trying to lightly cover the entire book, it is better to aim for a thorough understanding of selected topics.
- Chapter 1: Introduction (6 hours)
- Chapter 2: Linear systems (8 hours)
- Chapter 5: Applications I: planar phase spaces (2 hours on Sec. 5.1, 5.2)
- Chapter 3: Non-linear systems in the plane (6 hours on Sec. 3.1-3.5)
- Chapter 3: Non-linear systems in the plane (4 hours on Sec. 3.6-3.9)
- Chapter 5: Applications I (4 hours on Sec. 5.3, 5.4)
- Chapter 4: Flows on non-planar phase spaces (4 hours e.g. 4.1.1, 4.1.2, 4.2.3, 4.3.1)
- Section 4.4.1: Hamilton’s equations, with a supplement on celestial mechanics and the 3-body problem provided by the instructor (2 hours)
- Chapter 6: Applications II: non-planar phase spaces (2 hours e.g. 6.1, 6.3.1, 6.3.3)
- Tests and leeway (5 hours)
Notes:
- Sections 5.1 and 5.2 can be used to review for a test, since they emphasize the main points from Chapters 1 and 2.
- Section 3.5 (Stability of Fixed Points) can be supplemented with a treatment of LaSalle’s Invariance Principle, say from R. Redheffer “Differential Equations”.
- Section 5.4 on the Van der Pol oscillator should be supplemented with pages 513-515 of Redheffer’s text.
- Section 4.2.3 would benefit from the statement that if a number is irrational, then its multiples (mod 1) are dense in the unit interval. It follows that each trajectory with irrational slope has dense image in the torus.
- Use of differential equations software is encouraged as a way of helping students visualize the concepts. For example, Iode is a simple free package that can numerically solve autonomous systems in the plane (see www.math.uiuc.edu/iode/).
- Instead of teaching a traditional lecture course, one could insist that students read the text and then spend class time on discussion.
- The text of Arrowsmith & Place is at a good theoretical level, but does not always have enough examples or exercises. Students should be encouraged to consult other texts also, such as Boyce & DiPrima, Hirsch & Smale, Redheffer.
Last modified June 20, 2002