MATH 4541 Dynamics and Bifurcations
TR 1:35-2:55 Skiles 254
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
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
two weeks and grade 2 or 3 exercises among the one assigned.
The final grade will be based on the following rules: 45%
midterms, 20% HW. Curving will be done on the final result.
- 1.1 The Simplest Example.
- 1.2 The Logistic Population Model
- 1.3 Constant Harvesting and Bifurcations
- 1.4 Periodic Harvesting and Periodic Solution
- 1.5 Computing the Poincare Map
Exercises: Chapter 1: 2, 4, 8, 10, 13, 15
Solution set for the first HW.
- 2.1 Second_Order Differential Equation
- 2.2 Planar Systems
- 2.3 Preliminaries from Algebra
- 2.4 Planar Linear Systems
- 2.5 Eigenvalues and Eigenvectors
- 2.6 Solving Linear Systems
- 2.7 The linearity Principle
Chapter 2: 2, 3, 6, 7, 12, 14
- 3.1 Real Distinct Eigenvalues
- 3.2 Complex Eigenvalues
- 3.3 Repeated Eigenvalues
- 3.4 Changing Coordinates
Chapter 3: 1, 2, 5, 9, 12, 13
Chapter 4: 3, 4, 5, 6, 8
- 6.4 The Exponential of a Matrix
- Notes: The Evolution of the Area (to be posted soon)
Chapter 6: 12, 13, 14
Preparation Exercise for the midterm.
Solution set for the second HW.
Solution set for the Midterm.
- 7.1 Dynamical Systems
- 7.2 The Existence and Uniqueness Theorem
- 7.3 Continuous Dependence of Solution
- 7.4 The Variational Equation
- Notes: An apllication (to be posted soon)
Chapter 7: 2, 5, 7, 9
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.
- 8.2 Nonlinear Sinks and Sources
- 8.3 Saddles
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:
- 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
- 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.
- Simple oscillatory Phenomena and the Method of Averaging. See for example Capter V of "Ordinary Differential Equation" by Jack Hale.
- 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.
- 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.