##
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
- 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: 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).

### Presentation Schedule: