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:

• 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:

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:

Third week

Material covered:

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

Exercises:

Fourth week

Material covered:

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

Exercises:

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

Exercises:

• 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)

Exercises:

Eighth week

Material covered:

• 8.2 Nonlinear Sinks and Sources
  
• 8.3 Saddles

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.