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:

- 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, 15Solution set for the first HW.

**Second week**

Material covered:

- 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

Exercises:

Chapter 2: 2, 3, 6, 7, 12, 14**Third week**

Material covered:

- 3.1 Real Distinct Eigenvalues
- 3.2 Complex Eigenvalues
- 3.3 Repeated Eigenvalues
- 3.4 Changing Coordinates

Exercises:

Chapter 3: 1, 2, 5, 9, 12, 13**Fourth week**

Material covered:

- 4.1 The Trace Detrminant Plane
- 4.2 Dynamical Classification
- Notes: The comparison Theorem

Exercises:

Chapter 4: 3, 4, 5, 6, 8Fifth week

Material covered:

- 6.4 The Exponential of a Matrix
- Notes: The Evolution of the Area (to be posted soon)

Exercises:

Chapter 6: 12, 13, 14Preparation Exercise for the midterm.

Solution set for the second HW.

Solution set for the Midterm.

Makeup midterm.

Sixth week

Material covered:

- 7.1 Dynamical Systems
- 7.2 The Existence and Uniqueness Theorem
- 7.3 Continuous Dependence of Solution

Seventh week

Material covered:

- 7.4 The Variational Equation
- Notes: An apllication (to be posted soon)

Exercises:

Chapter 7: 2, 5, 7, 9Eighth week

Material covered:

- 8.2 Nonlinear Sinks and Sources
- 8.3 Saddles

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:

- 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.
- 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.