### 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:

• 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.

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:

Exercises:

Chapter  4: 3, 4, 5, 6, 8

Fifth 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, 14

Preparation 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, 9

Eighth week

Material covered:

• 8.2 Nonlinear Sinks and Sources