Math 6100
Dynamical Systems
Fall, 2009
| Instructor: |
Dr. Xiaoqiang Zhao, HH-2009, zhao@mun.ca |
| Time: |
Tuesday and Thursday, 2:00-3:15pm |
| Classroom: |
HH-3017 |
| Office Hours: |
Tu 10:30-12:00, Wed 13:00-14:30 |
| Course Website: |
http://www.math.mun.ca/~xzhao |
Text:
M. W. Hirsch, S. Smale and R. L., Devaney,
"Differential Equations, Dynamical Systems, and An
Introduction to Chaos", Elsevier, Academic Press, 2004.
Evaluation:
| Assignments |
50% |
| Project(written submission and oral presentation) |
50% |
Course Contents:
- Nonlinear Systems (Chapter 7)
- Dynamical Systems
- The Existence and Uniqueness Theorem
- Continuous Dependence of Solutions
- The Variational Equation
- Equilibria in Nonlinear Systems (Chapter 8)
- Some Illustrative Examples
- Nonlinear Sinks and Sources
- Saddles
- Stability
- Bifurcations
- Global Nonlinear Techniques (Chapter 9)
- Nullclines
- Stability of Equilibria
- Gradient Systems
- Hamiltonian Systems
- Closed Orbits and Limit Sets (Chapter 10)
- Limit Sets
- Local Sections and Flow Boxes
- The Poincare Map
- Monotone Sequences in Planar Dynamical Systems
- The Poincare-Bendixson Theorem
- Applications of Poincare-Bendixson
- The Lorenz System (Chapter 14)
- Introduction to the Lorenz System
- Elementary Properties of the Lorenz System
- The Lorenz Attractor
- A Model for the Lorenz Attractor
- The Chaotic Attractor
Assignments:
You should do your assignments independently. The assignments
will be collected before the class starts on the due date.
No late assignments are acceptable.
Assignment 1, Due date: Tuesday, Sept. 29
Exercises (Chapter 7): 2, 6, 7, 9
Assignment 2, Due date: Thursday, Oct. 8
Exercises (Chapter 8): 1, 3, 4, 9
Assignment 3, Due date: Thursday, Oct. 22
1. Prove that the omega limit set of a bounded entire
solution to the n-dimensional system x'=f(x) is compact
and invariant.
2. Exercises (Chapter 9): 3, 6, 10
Assignment 4, Due date: Thursday, Nov. 12
Exercises (Chapter 10): 2, 5, 7, 10, 14
Assignment 5, Due date: Thursday, Nov. 26
Exercise 8.11, Exercise 9.4, Exercise 9.11, Exercise 10.16
Project Topics:
1. The pendulum with constant forcing (Section 9.5).
2. Chemical reactions that oslillate (Section 10.7).
3. Competition and harvesting (Section 11.4).
4. Neurodynamics (Section 12.5).
Presentation Schedule:
2:00pm-5:00pm, Wednesday, Dec. 9, HH-3017.