In this first semester of a year long graduate course in differential equations, we shall focus on ordinary differential equations and dynamical systems. The second semester, Math 6420 taught by P. Bressloff, will emphasize partial differential equations. In this course, along with the Math 6420, we shall try to cover the syllabus for the qualifying exam in differential equations. Although some mathematical sophistication is required to take the course, and it moves at the blazing speed of a graduate course, I shall provide any background materials needed by the class. If you are unsure about background material, I EXPECT YOU TO ASK ME so I know what needs covering.

Outline

We shall follow Siders's text covering the behavior of solutions: existence and uniqueness, continuous dependence on data; and dynamical systems properties: long time existence, stability theory, Floquet theory, invariant manifolds and bifurcation theory. We shall discuss as many applications as we can.

I recently chose Sideris for the main text. Let us say that this text is "recommended," but you should own at least some of the better texts (Amann, Barriera-Valls, Chicone, Hartman, Kong, Perko, Schaeffer-Cain, Teschl) suitable for the course. The two other new books that I was trying to choose from included Kong, and Schaeffer-Cain. Siders is a no-nonsense book that proves all of the theorems and covers almost exactly Utah's course. Kong is a very readable, non-pretentious book aimed more at the first year student which omits some desirable material. Schaeffer-Cain is a chatty, amusing, readable but verbose treatment that has plenty of examples and exercises and a shallow learning curve, but doesn't cover some important material.

Topics include (depending on time):

• Introduction to ODE. Applications. Review of calculus.
• Linear systems and stability.
• Existence, uniqueness and continuity theorems.
• Qualitative theory, Lipunov stability, Limit sets and attractors.
• Applications to physical and biological systems. Charged particle, coupled pendula, planetary systems.
• Invariant manifolds and linearizations. Hartman-Grobman theorem.
• Planar flows. Poincaré-Bendixon theory.
• Periodic solutions and their stability.
• Sturm-Liouville Theory.
• Bifurcation Theory.
• Chaos.
• Perturbation Methods

Expected Learning Outcomes

• Local Theory
• Proof of the Local Existence and Uniqueness Theorem for ODE's
• Continuation of solutions
• Gronwall's Inequality
• Dependence of the solution on parameters
• Contraction Mapping Principle
• Linear Equations
• Linear systems with constant coefficients
• Jordan Normal Forms
• Matrix exponential and logarithm, Fundamental Solution
• Variation of Parameters Formula
• Floquet Theory for periodic equations
• Stability
• Liapunov Stability, Assymptotic Stability of solutions
• Liapunov Functions
• Proof of the Linearized Stability for Rest Points Theorem
• Grobman - Hartman Theorem
• Stable and Center Manifold Theorems
• Dynamical Systems
• Omega limit sets and limit cycles
• Poincaré-Bendixson Theorem
• Poincaré Map and stability of periodic orbits
• Bifurcation Theory
• Persistence of periodic orbits
• Normal forms for saddle-node, transcritical and pitchfork bifurcations
• Hopf Bifurcation
• Perturbation Methods

Grading

In addition, the student's performance will be reported to the Graduate Committee, which decides the continuation of financial support annually. Ultimately, the learning will also be measured by the Differential Equation Qualifying Examination.

Last updated: 7 - 13 - 17