MATH 4541 Dynamics and Bifurcations

Fall 2010

TR 1:35-2:55 Skiles 254

Professor Federico Bonetto

Office Hours: TR 3-4, Skiles 133B.
Grader Name: TBA
Grader Office Hours: TBA.

The course will mostly deal with the study of the long term behaviour of the solutions of planar differential equations bot linear and non-linear. 

The text that will be used is:

 Morris W. Hirsch, Stephen Smale and Robert L. Devaney Differential Equation, Dynamical Systems & An Introducyion to Chaos, 2nd ed., Elsevier

The syllabus can be found here.

There will be two midterm. The exercise listed are for HW collection. I will collect them every two weeks and grade 2 or 3 exercises among the one assigned. 

The final grade will be based on the following rules: 45% final, 35% midterms, 20% HW. Curving will be done on the final result.

First week

Material covered:


      Chapter  1: 2, 4, 8, 10, 13, 15

Solution set for the first HW.

Second week

Material covered:


      Chapter  2: 2, 3, 6, 7, 12, 14

Third week

Material covered:


      Chapter  3: 1, 2, 5, 9, 12, 13

Fourth week

Material covered:


      Chapter  4: 3, 4, 5, 6, 8

Fifth week

Material covered:


      Chapter  6: 12, 13, 14

Preparation Exercise for the midterm.

Solution set for the second HW.

Solution set for the Midterm.

Makeup midterm.

Sixth week

Material covered:
Seventh week

Material covered:


      Chapter  7: 2, 5, 7, 9

Eighth week

Material covered:

Structure of the Final Exam: You can choose an argument for a project involving differential equations and the material studied in class. Below are some ideas for project with some references. More will be added soon. The project should result in a written report of about 5 to 10 pages on what you read that show understanding of the mathematical ideas involved.
If you prefer not to develop a personal project, I will prepare a final exam (in class or take home depending on the number of requests).

Ideas for project:

  1. The Existence and Uniqueness Theorem: the Appendix of the books contains the classical proof of The Exitence and Uniqueness Theorem uning Picard Iteration. After learning that proof you can develop a proof of the same theorem based on the Euler methods to numerically solve a differential equation. Such a proof can be found in many numerical analysis book or you can discuss it with me. You can then extend your proof to obtain the Continuous Dependence Theorem.
  2. Stable and Unstable Manifold for Fixed point in dimension greater then 2: We have just proved the existence of a stable line for an hyperbolic fixed poind in 2 dimension. A similar theorem can be proved in higher dimension. See for example Chapter 4 of the "Ordinary Differential Equation with Application" by Carmen Chicone. 
  3. Simple oscillatory Phenomena and the Method of Averaging. See for example Capter V of "Ordinary Differential Equation" by Jack Hale.
  4. Keplerian two bodies problem and perturbations. See for example Sections 3.2.2, 3.2.3 and 3.2.4 of "Ordinary Differential Equation with Application" by Carmen Chicone.
  5. The Lotka-Volterra Equation: Basic properties of the equation. Extension to harvesting and 3 species.

Fianl Exam: here is the final exam. Please turn it in by Friday December 17th at noon. Better if you can email it. If you cannot email it, leave it in my mailbox and send me an email to let me know you dropped it in. The final is optional if you are preparing a project. If you are preparing a project, you can still turn in the final to improve your grade.