## Lecture Notes, Winter 2010

WARNING: downloading the lecture notes
and putting them under your
pillow at night will not help you learn the material.

If
you would like to have the movies that I showed, please come to see me
to copy them directly from my computer - most of them are very large.

Please
let me know if there are any mathematical errors remaining in these
notes.

Week 1:

**Preparation for class:**

- Preparation for Tuesday class: come and enjoy the show.
- Preparation for Thursday class: Read Strogatz Chapter 2.

**Lecture notes:**

- Lecture 1 (ppt presentation): Introductions
- Lecture 2 : Flows on the line

Week 2:

**Preparation for class:**

- Preparation for Tuesday class: Read Strogatz Chapter 3
- Preparation for Thursday class: TBA
- Lectures 3 and 4 : Bifurcations in 1D systems

Week 3:

- Preparation for Tuesday class:
- Preparation for Thursday class: Read Strogatz Chapter 5
- Lecture 5 : Example of population dynamics
- Lecture 6 : Introduction to 2D systems

Week 4:

- Preparation for Tuesday class: make sure you remember what an eigenvalue/eigenvector is, and practise finding eigenvalue/eigenvectors for 2D matrices.
- Preparation for Thursday class: download the ODEtemplate and examples from the Homework page and make sure you know how to use them.
- Lecture 7 : Linearization near a fixed point
- Lecture 8 : Linearization near a fixed point (part II)
- First inquiry : Numerical and analytical study of dynamical systems.

Week 5:

- Preparation for Tuesday class: finish up your inquiry on your particular system. You will be expected to make a 1-2 minute presentation on your system, and what you found. Also, if you haven't done it already, read Strogatz Chapter 5 and also 6.1-3
- Preparation for Thursday class: Read the rest of Strogatz Chapter 6
- Lecture 9 : Linearization near a fixed point: the general case.
- Lecture 10 : Example of applicaton: the damped Pendulum. Problems with linearization.

Week 6:

- Preparation for Tuesday class: Read Strogatz Chapter 6.5-7
- Preparation for Thursday class: Read Strogatz Chapter 7
- Lecture 11 : Structural stability. Conservative systems.
- Lecture 12 : Limit cycles (part 1).

Week 7:

- Preparation for Tuesday midterm: practise on previous year midterms!
- Preparation for Thursday class: work on the take-home!
- Lecture 13 : Limit cycles (part 2).

Week 8:

- Preparation for Tuesday class: Read Strogatz Chapter 10.1-10.4
- Preparation for Thursday class: Read Strotatz Chapter 10.5 and 11
- Lecture 14 and 15 : Introduction to discrete systems.

Week 9:

- As you may have noticed, because we have taken some delay in the class schedule, I have had to skip a few things and go in less depth than I would like on others. Please read the textbook for details, or come and ask me.
- Lecture 16 : Introduction to discrete systems. Lyapunov exponents as measure of chaos. From discrete systems to fractals.
- Preparation for Thursday class: Bring a laptop!
- Inquiry: The Rossler attractor and the Lorenz map. See Exams page for answers.

Week 10:

- Lecture 17 : 2D discrete systems. Behavior of dissipative vs. non-dissipative maps.
- Lecture 18 : Fundamental examples of dynamical systems: Synchronized Oscillations
- If you are interested : Fundamental examples of dynamical systems: The square Map.

Final's Week:

- Schedule for the presentations:

- 2:00 - 2:15: Ariel Kellison: The dynamics of nonlinear convection
- 2:15 - 2:30: Claire Walton: The cubic map
- 2:30 - 2:45: Katelyn White: Noisy dynamos
- 2:45 - 3:00: Jen Holt: Particle trapping in vortices
- 3:00 - 3:15: Curtis Andrus: the Newton Raphson method
- 3:15 - 3:30: Ross Anderson: Chaotic skiing
- 3:30 - 3:45: Justin Yeakel: Dynamics of a food web
- 3:45 - 4:00: Chris Phelps: The restricted 3-body problem