Math 634: Theory of Ordinary Differential Equations

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Catalog Information

Title

Theory of Ordinary Differential Equations.

Credit Hours

3

Prerequisite

Math 316, 334.

Description

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

Desired Learning Outcomes

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

Prerequisites

Minimal learning outcomes

  1. Solutions to ordinary differential equations
    • Existence of solutions
    • Uniqueness of solutions
    • Continuation of solutions
    • Gronwall’s inequality
    • Dependence on parameters
    • Contraction mapping principle
  2. Linear differential equations
    • Linear systems with constant coefficients
    • Jordan Normal Form
    • Fundamental solutions
    • Variation of constants formula
    • Floquet Theory for periodic solutions
  3. Stability and instability
    • Stability and asymptotic stability
    • Lyapunov functions
    • Bifurcations
  4. Poincare-Bendixson Theory
    • Invariant sets
    • Omega limit sets
    • Limit cycles

Additional topics

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.

Courses for which this course is prerequisite

Math 635