Math 525

  Ordinary

Differential Equations II

University of Alberta
Mathematical and Statistical Sciences
Dr. Thomas Hillen
492-3395, thillen@ualberta.ca

Cab 575

1. Assignment 1
2. Assignment 2

Syllabus: 

In this course we will study asymptotics of ordinary differential equations and boundary value problems. The Poincare-Bendixson theory has been covered in Math 524. We begin with Floquet theory for the stability of periodic attractors.  Additional material covers the theory of dynamical systems and differential equations in Banach spaces. The concepts of stability and bifurcations can be generalized from ODEs to PDEs. We will  systematically derive a theory of finite dimensional  compact global attractors, and we will investigate two examples in detail: the Navier-Stokes equations and reaction-diffusion equations.

Texts:

• J.C. Robinson.  Infinite-Dimensional Dynamical Systems. Cambridge University Press, 2001.

• R. Temam.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, 1988
• O.A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge 1991.
• M.W. Hirsh, S. Smale.  Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, 1974.
• L. Perko. Differential Equations and Dynamical Systems. Springer, 3rd ed., 2001
• A.V. Babin, M.I. Vishik, Attractors of Evolution Equations. North-Holland, 1992

  Grading:   

   Homework 70%, in class presentation 30%

  Contact:

   Dr. Thomas Hillen, 492-3395, thillen@ualberta.ca
  office hours: after calss 9:50-10:30, or by appointment,   CAB 575.
  

  Policies:

  Policy about course outlines can be found in Section 23.4(2) of the University Calendar.

  Academic honesty:

  The University of Alberta is committed to the highest standards of academic integrity and honesty. Students are expected to be familiar with these standards regarding academic honesty and to uphold the policies of the University in this respect. Students are particularly urged to familiarize themselves with the provisions of the Code of Student Behavior (online at www.ualberta.ca/secretariat/appeals.htm) and avoid any behavior which could potentially result in suspicions of cheating, plagiarism, misinterpretation of facts and/or participation in an offence. Academic dishonesty is a serious offence and can result in suspension or expulsion from the University.

Course Notes 

1. ODE's
2. Discrete Dynamical Systems
3. Connection of Discrete and Continuous
4. Abel's formula and the Wronskian
5. Floquet Theory
6. Periodic Attractors
7. The Lorenz Equations

1. Banach Spaces
2. Mollifiers
3. Some useful integral estimates
4. Hilbert Spaces
5. Linear Operators 
6. Dual Spaces and Weak Convergence
7. Sobolev Spaces
8. Banach-Space Valued Functions

1. Modelling
2. Basic Assumptions
3. Weak Solutions (Galerkin Approximation)
4. Strong Solutions

1. Pressure and Fluid Velocity
2. The Stokes Operator
3. Weak Formulation of the N-S eq.
4. Weak Solutions
5. Uniqueness in 2-D
6. Strong Solutions 

1. Dissipation, Limit Sets and Attractors
2. Structure of the Attractor 
3. Shadowing
4. Continuous Dependence on Parameters

1. Absorbing Sets and the Attractor
2. Injectivity
3. A Lyapunov Function
4. The Chaffee-Infante Equation

1. Global Attractor
2. Injectivity

1. Fractal and Hausdorff Dimesnion
2. Evolution of n-Dimensional Volumes
3. Reaction-Diffusion Equations
4. Navier-Stokes equations