## Math 6100 Dynamical Systems

### Fall, 2017

 Instructor: Dr. Xiaoqiang Zhao, HH-2009, zhao@mun.ca Time: Tuesday and Thursday, 2:00-3:15pm Classroom: HH-3013 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
• Stability
• Bifurcations
• Global Nonlinear Techniques (Chapter 9)
• Nullclines
• Stability of Equilibria
• 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: Thursday, Sept. 28

Exercises (Chapter 7): 2, 6, 7, 9

#### Assignment 2, Due date: Tuesday, Oct. 10

Exercises (Chapter 8): 1, 3, 4, 9

#### Assignment 3, Due date: Thursday, Oct. 19

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: Tuesday, Oct. 31

Exercise 8.11, Exercise 9.4, Exercise 9.11, Exercise 10.2, Exercise 10.5

#### Assignment 5, Due date: Thursday, Nov. 16

Exercises (Chapter 10): 7, 10, 11, 14, 16

### Project Topics:

1. The pendulum with constant forcing (Section 9.5).

2. Chemical reactions that oscillate (Section 10.7).

3. Competition and harvesting (Section 11.4).